Run Death House
Death House serves much better as an introduction to CoS over Phandelver. Phandelver is a great adventure, and I started as a player with it, but it would serve you better later if you want to run a classic D&D adventure, with less gothic horror elements. I'm DMing Curse of Strahd now and started my group with Death House.
Death House will teach you certain DM skills, and it will reinforce the skills necessary for the PCs to survive Curse of Strahd. Here's my justification:
Introduce with Roleplay and dread
Curse of Strahd is a roleplay-heavy campaign — there are lots of situations you don't want to bash your way through. Additionally, Death House will, if played correctly, teach you as DM to foreshadow threats and create a sense of dread that will help with the rest of the campaign on both sides of the table.
For example, the desolate streets of Barovia, with only a single house on the outskirts lit up, with smoke pouring out of the chimneys will draw the players in, but also give them the eerie feeling something is wrong. You'll have to play the children correctly to bait the players in, but you can always use the mists to railroad them as necessary. Once they're in the house, you'll be able to hook them in.
This introduction is much more in-line with the tone and nature of the rest of the campaign.
Take time to describe the environment
This is critical with Death House, but less emphasized with Phandelver. With Death House, you'll need to take time in each room, describing the blazing hearths, the strange wolf-helm armor, everything. It's all important, because it also teaches the players to pay attention to the environment. When
the blazing hearths pour out poison smoke in the 2nd phase,
they'll get a good idea of how Barovia works. You'll have plenty of chances to have them roll Investigation, Perception, and Insight during the buildup, and therefore teach them some fundamental rules, and fundamental survival skills for later.
Toss a bit of combat in after
The first fight will likely occur later,
on the 3rd floor.
This will be after the players have had the run of the house for a bit, and will give you plenty of time to switch from warm, but empty house, to
the dusty, old, decrepit 3rd floor.
Using that buildup, when something bad happens like
the armored statue punching a player in the face,
the players will be alert to such changes. It's also an easy fight to run, but will introduce damage resistance and basic combat rules.
Phandelver starts you off with a goblin ambush, which is fine, but Curse of Strahd places the emphasis on the fact that the players can walk into dangerous areas — Phandelver's goblin ambush is more of a device used to spur the plot along (perhaps much like the mists). Death House will give you a chance to explore first. Having the combat after emphasizes where the priorities should be in this campaign, I think.
Ramp up the danger
Death House is notorious for how lethal it can get — your players can absolutely TPK against
the Shadows, or the Shambling Mound in the basement.
However, much of the campaign is like this — you'll get to make the decision of how hard to beat them over the head with their impending doom, or how to fail forward. You'll learn from your players what they expect out of a campaign with this kind of tone, and it'll get everyone on the same basis. Starting with Phandelver, I think you stand the danger of letting them be the big damn heroes and then taking that away from them.
Consider not worrying about it and just continuing on. At Adventurer's League, tables are divided by tiers, and while new players (or players with new characters) start at level 1, that same table can have players from level 1 to 4. I know you're not doing Aventurer's League, but the same "tier" concept still applies — see chapter 1 in the DMG.
In fact, I've specifically played (parts of) Lost Mine of Phandelver both as a 1st-level character when there were also 4th-level characters at the table, and as a 4th-level character when there were others at 1st level. In both cases, it was fine. The higher-level characters took more risk and did some covering for the lower-level ones, but (especially once over the jump from 1st to 2nd), there wasn't even much of a real-world issue of the higher-level characters taking up too much game spotlight.
You mention that the module is meant for 1st level characters, but actually it's meant to take characters from 1st to 5th. Think of it as a first tier adventure. Starting at 4th certainly makes things easier, but... it'll all work out. Particularly, consider that it takes 2700 experience points to get to 4th level, but 6500 (that is, 3800 more) to get to 5th. Your other players will catch up, and everything will be fine.
Best Answer
This answer contains spoilers for the Death House, and is based on my personal experience playing it as well as the Curse of Strahd source itself.
First things first, a "dedicated healer" is something that is seldom actually seen or used in DnD 5e. The game lends poor support to characters intending to use most of their time healing other characters, and healing output from spells is generally paltry compared to the amount of damage per turn. The best way to keep your friends alive is, in most cases, to help defeat their enemies quickly instead of trying to repair the damage they do on the fly.
That said, healing magic or other powers like the Healer feat are very useful for rescuing characters from zero hit points --- even a single hit point is enough to give them full actions next turn and prevent them from having to make death saves.
But Death House is not quite your average adventure. Yet another reminder: spoilers ahoy.
The Death House is extremely deadly, assuming your players make the wrong --- yet completely sensible --- choices. Combat encounters, usually being the most dangerous part, are actually not the worst thing here: rooms filled with poisonous fog and doors turned into spinning scythe blades will very quickly wear a party to the end of their resources, unless they are exceedingly lucky with their rolls. Loss of health is not easily avoidable even with conservative gameplay, and rescuing unconscious characters quickly becomes a liability without fast healing in the conditions of the house.
To illustrate: our party went in with full resources. We had a Paladin for healing, and my Monk had the Healer feat and a Healer's kit with me. We emerged with all slots expended, the entire Healer's kit used up, with two characters at zero HP, the other two at a single HP, and this was with me and the Paladin squeezing everything we could out of our respective healing resources in a manner many would consider cheesing. Death House is that deadly.
Suggestions to alleviate the deadliness
The house is not at all that dangerous if your players pick the right choices. Since I've warned you already about spoilers, I'll state it up front: to avoid the house turning against the PCs, a creature has to die at the sanctum in the basement. Being open about the deadliness of the adventure can prime your players to make the right choice there and accept the consequence of losing one of their own. While not obvious, the sacrifice does not have to be humanoid in nature: any creature will do. If you're feeling particularly merciful, give them a henchman NPC to tag along to sacrifice, after which the party can leave the house unhindered, or some animal they can capture for that purpose.
For a less merciful alternative, you can fudge the rules of the house slightly. My Curse of Strahd GM suggested that the house should be appeased when a PC dies, even not in the sanctum. This will preserve the feeling of deadliness, but politely spares your players of the unsavory decision on who to sacrifice and gives them a shot at leaving everyone alive. You can of course also extend the players' capabilities to some extent by giving them health potions, but this is a bit fickle alternative: it might happen that the PC carrying the potions turns into a single-point-of-failure whose unconsciousness dooms the whole party.
Overall, whether or not this kind and extent of deadliness is desirable is up to you and your party. The best you can do is warn them in advance, really, that this is not going to be their average heroic romp but rather a nightmarish situation their characters will be lucky to survive and that some PC deaths are the expected outcome. If you or your players want that heroic romp instead, I suggest not playing Death House with them.