Risa

You are right on the Thermians saying they're the last of their kind but they say that after Tim Allen has bundled the first meeting. It's vaguely implied that they're the last of their kind because of what Tim Allen did.

I remember that now that you say it.

So I think it really depends on whether Mathesar represents continuity of government in Thermian law. If there's no continuity of government, then at that point they are non-state actors, paramilitary forces in exile.

For sake of argument, let's say we're meeting them before their planet is destroyed.

Well we did see the Enterprise involve itself in various skirmishes without too much oversight or approval so any approval needed might just be a quick message to Starfleet.

That really depends on the nature of the conflict and which version of the Enterprise we're talking about.

I haven't seen much of TOS so I don't know what Kirk would do, but if we're talking TNG era I think that Starfleet would be pretty hesitant to get involved in a conflict they didn't start.

We also have to consider where in the galaxy we're meeting the Thermians.

If it's in the Alpha quadrant near to Federation space, then military intervention isn't out of the question, especially if they've expressed interest in joining the Federation. If they're in Klingon or Romulan or Breen space, or the Neutral Zone, then Starfleet getting involved could be a real problem. They may still render humanitarian aid and even clandestine military aid, but they're going to have to tread very carefully.

If they're in the Gamma quadrant through the wormhole, then Starfleet may be hesitant to commit any significant assets, even if they want to help. It'd especially be an issue once contact with the Dominion is made. Sarris could even be working with them as an enforcer if it's a remote part of their space they don't really care about.

If they're in the Delta quadrant asking Janeway for help, which is the easiest way to slot this conflict into a TNG era Milky Way from a narrative perspective, then it really depends on the size of Sarris' forces and whether Janeway thinks she can win against him. If it's an even fight, she might see aiding the Thermians and gaining an ally in the region as worth the risk. If Sarris is too dangerous she might just say "sorry" and move on, maybe leaving behind some medical supplies and assisting with some repairs and upgrades. She'd be really torn about it though, and some of the crew would complain or even clandestinely join the Thermians to help. It'd make for a decent bottle episode.

Wait, what? I don't remember that being implied at all. That would be incredibly dark for such a light-hearted comedy.

The Thermians were fighting Sarris long before they brought in Tim Allen's character.