Take the Martial Adept feat

The key is in the wording of the feat and the Battle Master maneuvers section. The first bullet of Martial Adept says (PHB, p. 168, emphasis mine):

You learn two maneuvers of your choice from among those available to the Battle Master archetype in the fighter class. [...]

The second paragraph of the Battle Master maneuvers section, quoted in the question, says (PHB, p. 73, emphasis mine):

You learn two additional maneuvers of your choice at 7th, 10th, and 15th level. Each time you learn new maneuvers, you can also replace one maneuver you know with a different one.

Since Martial Adept lets you learn additional maneuvers, the Battle Master feature allows you to switch one of your previously-selected maneuvers with another one. The feat also grants two additional maneuvers and one additional superiority die. No additional fighter levels required.

Maybe, maybe not

This party make up is far from ideal, but it might be doable.

How doable will depend on how much guidance they accept from you, as the DM.

The main problems

Healing

You know this already, there is very little healing in the party, and none at level 1.

There are a couple ways to mitigate this, however.

Give them healing potions.

Have the Ranger pick up Goodberry as a spell at 2nd level, and probably cure wounds. Make sure they cast Goodberry and share the berries with everyone, and make sure they understand that, most of the time, they will want to use those just on unconscious party members. Otherwise, there's going to be a lot of waiting 1d4 hours for someone to wake up after being stabilized.

Make sure the Goliath understands and uses their Stone's Endurance ability. It basically acts like healing, as they can use their reaction to reduce the incoming damage. This kinda acts like a once per rest self only Healing Word. Not great, but better than nothing.

Armor Class

While the lack of healing is bad, the potential lack of armor is worse. I DM'd this campaign for 2 characters: Dwarf Cleric (18 AC) and Halfling Rogue (14 AC).

That 4 difference is AC was huge, and the Rogue went down repeatedly. The cleric was able to cast Healing Word constantly to bring her back up, but that won't be an option for your party.

On the plus side, it is possible for each of your players to get their character's AC to 16, even at 1st level. This again requires that they be willing to follow your guidance.

Barbarian Goliath

Unarmored defense will provide 16 AC if there is at least 16 DEX and 16 CON for this character -- both easy to get with a Goliath.

Tiefling Ranger

Scale mail with 14 DEX will provide 16 AC. However, they will have disadvantage on stealth. The alternative (using the class's starting equipment) is leather armor, which with 14 DEX is only going to provide 13 AC. The Tiefling can't do any better than +2 AC from DEX unless you are rolling stats and they get lucky.

Elf Sorcerer

Draconic bloodline with 16 DEX provides 16 AC as well. As an elf, this is pretty easy to achieve.

New players

Your players are new, which further increases the difficulty. When I ran this campaign, both my characters were new to D&D (as was I), but myself and one of the players had at least seen it played before, while the second player had no clue what to do.

Your players won't really understand their options, or what is effective, or what to look out for unless they receive a lot of guidance early on.

However, my players were also very careful (in general), so if your players want to play more reckless characters, they won't survive with the current party make up.

The first encounter

While preparing for my own running of this campaign (which we just wrapped up) I learned that the first encounter is notorious for TPKs, or at the very least some characters outright dying and their brother with identical stats and just a different name joining the party to replace them soon after.

Things are going to be even worse for your party, for all the reasons I outlined above. I watched a video of several experienced players (and an experienced DM) and one newbie. During the first encounter, the newbie wasn't even sure what she could do, and ended her first turn in the middle of the road with her action unused. In a group of 5 they were able to cover for her basically doing nothing in the battle, but in a group of 3 that is going to get you slaughtered.

Fortunately, there is a way to survive it and help the player's get used to the game, without being TPK'd on their first fight.

Make the first encounter a handholding tutorial

As my Rogue's player was brand new, I literally walked her through what to do during the first encounter. I taught her about stealth, having advantage when being unseen, sneak attack damage, and more.

I also had one of the melee goblins asleep, and so didn't participate in the battle.

By treating the first encounter as a tutorial, the players were able to both learn how to play and were able to easily handle the goblin ambush.

The players got to learn about traps from that first path they used to find the cave, and then I let them use what they learned in the first encounter on the 2 goblins outside the cave. The Rogue dropped unconscious twice during that fight, but they prevailed.

They then took a long rest and leveled up to level 2 before entering the cave.

I strongly recommend you get them up to level 2 at this point, as well, if you don't make any other adjustments.

From this point, if they play smartly and cautiously, I think they will be okay. You may have to accelerate their levels slightly. I eventually settled on basically doubling my players' characters' levels, as well as maxing out their HP (and the HP of all enemies) to smooth out the combat, but I don't think you will need that much with 3 players.

What if any of the above doesn't apply

However, if the players want to be reckless (and if they have the most fun that way, you should let them be reckless), or don't want to follow your guidance on building their characters (they are their characters, after all), or bristle at the idea of a tutorial to start with, then you'll need to help them in other ways.

Life Domain Cleric

One of the best ways is to just have an NPC Life Domain Cleric along for the ride. High AC, great healing, powerful undead control. All things you'll need and that will make things go a LOT smoother.

Boost their levels a lot

Alternatively, you can do as I mentioned above and boost the characters to essentially twice their normal level. Having just completed the campaign, with the final fights happening at levels 8 and 9, it felt like it worked pretty well. (That wasn't an arbitrary doubling, I actually computed the expected XP difficulty of encounters for a party of 4 at the adventure's intended levels, compared them to a party of 2, and it roughly worked out to doubling the party of 2's levels to make them line up nicely.)

The extra levels will offset their deficits, and help cover for the aggressive play style that they seem to be going for. They may be overpowered, but it is a lot easier to up the difficulty of an encounter than to reduce it. Also, if they enjoy power fantasies (my players did) then they won't complain.