Use your character's backgrounds to build intrinsic motivation for your players' characters.
If you're looking at the LMoP adventure, you hopefully have the pre-built characters handy. If you look at their backgrounds, you can see that for every character, there is some kind of intrinsic motivation built into their backgrounds for at least one of the elements of the adventure.
This provides the characters with clear motivation for why they should care about both the mission and the results of the mission.
You can do this in a custom setting in a couple of different ways. First, you could assign different background elements to your players, this may be a bit of a deal breaker for some players, but I've had good luck with it using a pre-built adventure and giving my players the option of which of 10 or so to select (Horde of the Dragon Queen provides 10 different optional background elements).
However, the best thing you can do is to pick up plot elements from your characters' backgrounds. Have them write a short paragraph or even just be super detailed when they are building their BIFT part of their background so that you can mine that for details and NPCs and other motivations.
If you're looking to provide intrinsic motivation, having it worked into your characters' backgrounds is the best and easiest way to do it.
Contains spoilers about LMoP. As a background, I've been playing it for a few weeks with newbie players, although I'm not a newbie DM, so that part of your question I won't be able to answer. About the broader question, I think it's unanswerable - it depends entirely on the adventure. Cos or HotDQ are probably impossible to one shot, for example.
First Option - Run only part of the adventure.
If you don't want to run the whole adventure, but only a part of it as a one shot, it should be fine. Some one shots let hooks for future adventures. One way to one shot LMoP, giving some sense of ending while letting some mystery is to put Gundren with Sildar in the first cave. You rescue them, deliver them safely in Phandalin and that's it. They thank you, give you the reward and everyone gets back to their lives. They are still secretive about what's happening. If the players decide to keep playing and insist on learning about Gundren and the Cave, he tells them about the map - but the map itself was lost when he was kidnapped, and he can't remember exactly where the cave is (or simply doesn't know - he didn't look carefully at the map yet - or something along these lines). The map then can be found where normally Gundren would be.
Thundertree and Cragmaw castle can also be run as one shots, if you would prefer your players starting at a higher level and defeating some harder challenges.
Personally, I don't like the idea of one shoting WEC because there are too many plot hooks before it and it would feel a lot awkward for me to run it without the background.
Second Option - Run the whole adventure, cutting off what you can.
If you are, instead, interested in running the whole story, from the hook at Neverwinter (or other hooks you might have used) to the end, in the Wave Echo Cave, I'll first ask (and answer) honestly: can you?
Probably not.
If you want to run the whole (main) adventure (I mean just the main plot), i.e., from the hook to the Wave Echo Cave, it will almost certainly take more than two sessions.
Essentially, each chapter from LMoP can be a one shot. The first chapter is supposed to be a one shot, while the others should take 1 or 2 average sessions. With new players, that usually ends up increasing some time, especially if they are undecisive or have to check the books constantly - there are other questions on how to solve these problems, though (usually "How to speed up a game" or similar)
If each chapter should take (at least) one session, it's unlikely you can rush four chapters.
Things you can cut off, though
I will write about things that either my players skipped and felt no regrets for it or things they (or I) wished were skipped, also things that are explicitly stated as optional from the book itself.
First Chapter
The Goblin Ambush is a good first fight - it presents battle concepts like hiding, advantage and usually teaches someone that they shouldn't be wandering alone and getting attacked by 4 goblins at once and almost dying.
The Cragmaw Hideout/Cave is a loaded dungeon crawling, though. We, as a group, found this too much for a first session, tbh. There was a decent amount of roleplaying with the goblin interrogation and talking with Sildar, but other than that, it's lot of fights, some of them skippable but some players or PCs won't skip it. Myself, playing again I would cut the whole room with Yemrik and Sildar and put Sildar where the Bugbear (Klarg) is. It decreases some time of "WHAT PATH SHOULD WE CHOOSE?" and makes it considerably faster.
Note that skipping this part of the dungeon is common for players that rush it and led to some questions here about it, e.g. My players didn't rescue Sildar, what now?
Important alteration: make the remaining encounters a little harder and give considerably more experience.
Second Chapter
If you are going to rush, the third chapter is, as a whole, optional and skippable. This means most NPCs giving side quests here are also skippable. Other than that, it depends on the players rushing the Redbrand Hideout. My players did it (the question pretty much explains how to rush it) and it was really, really fast. If you want to force this rush, it's essentially cutting rooms 1-6 out. Do I recommend doing it? Not really. If you want to make sure they will be able to quickly proceed even if they kill Glasstaff, you can add the location of the Wave Echo Cave in the letter that Black Spider sent to Iarno.
Again, make the whole dungeon a little harder and give considerably more experience. By the end of this dungeon, they should be fourth level, which probably needs steroids for it to be possible (as they probably entered here with around ~700-900 XP and they need 2700 to get to 4).
Third Chapter
Skip everything except for the random encounters on their way to Wave Echo Cave. Depending on how fast you went through the first two chapters, you might be able to run the Cragmaw Castle. Sildar might have found it (instead of asking you to find and get rid of the goblins, he found it and get rid of them). Again, this depends on how fast you ran through the first two chapters. Note that Gundren is here. If you don't run it, either move it back to the Redbrand Hideout or to the Wave Echo Cave, depending on if you want Iarno/the letter to give the location of the cave or Gundren himself.
For reference, the problem with skipping this is that it's the chapter that gives the most amount of XP before the 4th chapter, BY FAR. I've made a table with how much XP each side quest gives along the numbers for the two chapters. Note that the first two chapters give you a total of 4200 XP (1050/character in a 4 PCs party), while the 3rd chapter alone gives you 10600 XP. While only 10800 XP is needed for the party to reach 4th level, it still means you need to somehow compensate for 6600 XP.
Another possibility is to tone down the encounters in WEC for 3rd level, instead of 4th.
Fourth Chapter
Hopefully, you somehow got them to 4th level (or you weakened the encounters here so they are beatable by a 3rd level party with the same challenge as it would originally for a 4th level). Also hopefully, with all these rushs in the beginning, you will have time to complete the whole cave.
With these changes, you should still be able to run through the main plotline - Gundren and Sildar were kidnapped, you rescue Sildar first and then Gundren, discovering about an evil network of Goblins (only in the Cave though) and bandits (Redbrand hideout) and defeating the mastermind (Black Spider) behind them in the final battle stage (Wave Echo Cave).
Honestly I would still give two sessions to this rushed version - one for the ambush, cave and the hideout and another for the WEC. So, TL;DR: This version takes at least two sessions, and if you party doesn't rush it, probably more. This version also cuts essentially everything that is barely skippable, so I don't think you can reduce it further.
Best Answer
I'm not familiar with the adventure path you're using, so I can't speak to it specifically, but I can give some general advice.
There are, generally, four ways to handle this.
As you mentioned, is let them walk in and die. They'll, hopefully, learn from their mistakes and be more prepared in the future. A lot of people like this kind of play and it's perfectly valid, so long as they have a way to gauge the strength of the area they're going into or some warning that they might be in over their heads.
However, some players don't like having their characters die and some groups would rather not have to deal with the discontinuity. It can also be a bit immersion breaking for games that are more focused on story, rather than just tactical and mechanical prows. You said that you have younger players and they don't like their characters dying, so it might be best to avoid this method.
Come out and warn them outside of the game. "Just a heads up, the area you're going into is meant for higher level characters, so you might want to come back later." You should still give them the choice to go in, if they want to, but then the decision will be entirely on their heads and they'll also be able to react appropriately.
This method has the lowest chance of ending in tears and the best chance of ending in success, simply because it's straight forward and gives the players the information and opportunity they need to make an informed decision. That said, it's also the most meta-gamey and that can be a turn off for some.
You can also edit the campaign so they simply can't get in before they're ready. You said there's a banshee in the canon version that can tell them what they need? Ok, well, in your version she doesn't know. Simple as that. If they've never played the adventure before (and haven't read ahead) then they'll never know they missed anything. They won't learn anything, but they're guaranteed not to get to the cave too early.
Don't worry about it. If they get to the cave early, make sure there are some hints that they might be getting in over their heads, but let them go. Then if and when they get into a fight they can't win, let them run. You can also arrange the fight to minimize the chance of them dying, without going easy on them, by doing things like having the monsters prioritize still standing characters over fallen (but still living characters) and active combatants over characters that are trying to evacuate the fallen. If there's something about the cave that means they can't exit it once they enter, you could put a comparable fight (maybe a little bit stronger than average) at it's entrance to serve the same purpose.
This method is the most game friendly, but it also has the highest chance of someone dying (baring the one where you just let them all die).
Personally, I prefer the fourth method, since it's the most RP friendly and knowing when and how to retreat in an RPG is a good skill. I don't think the players should always win, but by the same token defeat shouldn't default to them dying. There's also the possibility that good tactics will make up for the level difference, in which case they'll feel really good about overcoming a more difficult challenge. Even if they retreat, getting stronger and coming back to beat someone who's already beaten makes for a great sense of accomplishment.
That said, if you're really worried about them dying and you think they won't handle it well, #2 and #3 will serve you quite well.