taking barbarian 1 just for the rage is really old. 14 years old truth be told. Good idea? Maybe. But delaying all ability bumps traded for partial damage reduction.
Dragon fire is not reduced by the barbarian rage damage reduction. Also a dragon can just fly away and come back later when the barbarian has clamed down.
Good? Certainly. Overpowered? Not too much. Maybe raging should prevent the use of heavy armor... but most probably dipping barbarian is a fair trade. Being one level later at everything is annoying. It could be just that level where you fight the dragon when you are one attack behind or when you wish you had indomitable to resist the fire breath.
I am more worried about dipping moon druid level 2 and will certainly rule that at lvl 2 you only be able to change into a 1/2 CR animal.
This is kind of like saying that multiclassing itself results in being annoying because everything is one level later.
Let's take level 9 fighter vs. level 8 fighter level 1 barbarian.
The level 9 fighter has indomitable 1/day. The other has rage 2/day.
HD 8 Young Green Dragon:
Breath 42 damage, DC 14 Con save, recharge 5-6
Claw/Claw/Bite: +7 to hit, 11/11/22
One does average 42 (30.5 DPR) in a single round, the other does average 44 (17 DPR) in a single round.
The breath has a recharge and can typically only be done 1 round in 3.
So regardless of how many rounds the dragon breathes, indomitable helps at most once. If it works, it stops ~21 damage. If it does not work, it does
nothing. In this example, on average it stops 55%*21 = 11.5 DPR points of damage.
A round where either fighter is hit with claw/claw/bite, it stops 6+6+8=20 damage (or more with criticals, on average 8 DPR for a round). And it does it for the entire encounter. Every successful hit.
One possible entire encounter of Indomitable saving just a little more average damage (11.5) as Rage does in a single round (8 DPR).
As for your "dragon can just fly away" comment, doubtful. In the 136 hit point Young Green Dragon example here, even in a single round, the PCs are probably going to drop 40 to 60 points of damage on him at 9th level, especially with casters throwing out their big guns. And as he flies away, unless he is out of sight, his 80 speed is going to allow the PCs to fire off another round of ranged spells and attacks. A dragon would be stupid to come back being half wounded or so. Granted, in examples where the Dragon is higher CR and is legendary compared to the party, the party does less damage and he does more, but even so, at 9th level, PCs have a ton of different offensive options to just nail a Dragon. Even a tougher one would at least be partially injured and the players are now aware of his existence. That option seems subpar for a Dragon unless he just rules the encounter and if he is that tough, there is no need for him to fly away and come back later. The fighter's resist isn't going to matter.
So no, I'm not seeing where Indomitable is so amazing. Once per day, it may or may not save from some status effect which is good. But this is only for 1 level. About 10 encounters total (according to the XP guideline which I think are very skewed low). At level 10, both fighters have it.
And the resist works day after day, 2 fights max per day for every level starting at 2 (assuming the player goes Fighter / Barb and not Barb / Fighter at levels 1 and 2).
I do not see anything on the Fighter's list at any level where I say "Boy, I'd really want that instead of resist melee (or ranged weapon) for one more level". Nothing on the list seems that good, although situationally, they might be. Even the Champion's Survivor does not typically save as much damage per encounter (since fighters might not go to half damage in many fights and while above half damage, it stops) and how often do people play at level 18?
From a party resource POV, this just means that the healers can save about 2 cures / tier / day that do not have to be cast on the fighter (the guy getting melee attacked the most), but can be shared elsewhere. Even with good AC, being up front, the fighter is going to be hit a lot every day unless the encounters are easy (or unless he is an archer).