Barbarian doesn't have spells.
The rest of this post is appreciated, however - I'll run through point-by-point on suggestions.
It might be because I'm unfamiliar with the eidolon mutations but it seems to not have a flat or close to flat rate of advancement: there are levels that are clearly huge jumps in power ability
That was an issue I didn't know how to solve, unfortunately. Although now that I look again, I might have some ideas that would fix this, and I think I missed a major point from the archetype's base features.
and there isn't anything that limits what you can take as a evolution (which I know people complain is an issue of eidolons) as for when or major perquisites.
Hmm, so I should just rewrite the whole evolution list for genetics? Time-consuming, but easily doable.
I like the ideas behind each dragon type and where it takes you, but it also seems you gain nothing without being in those forms so you have to be a raging dragon to be stealthy as a linnorm or charismatic as an imperial dragon.
Yeah, didn't think about that. Hmph, back to the drawing board I guess!
Since the barbarian is also going through a drastic change, not just going super saiyne (even then it still took goku 27 episodes), I would make the action a standard that provokes and maybe even a full round action; this is also clearly much more powerful than even a double dragon disciple which means that it should take some amount of time to activate.
I'm not sure it should take THAT much time, but maybe I could make a ruling on armor needing an enchantment to stay on. (There's an enchantment that allows a druid to keep their armor while wild shaping, right?)
Maybe a strategy of "once its up, you'll want to keep it up and be extra careful once it drops" could be implied by a longer startup time where you are completely vulnerable (for a couple rounds, but it doesn't eat your round usage until you have fully transformed) and once you drop it, you are exhausted for double the rounds you used it, with the level 17 ability making it only fatigued.
That seems a little extreme, but I'll look into that. I don't want to gimp this thing, but I don't want it too overpowered either.
Again, thank you!