First let me say that I certainly am no "pro" in this game. Furthermore I only played TEC so far and for the first time picked another race.
I use the 7-DeadlySins-Mod and picked the Atlantians for my race, but I guess for those who dont use that mod, you could compare it with the Vasari in the standard game for the "problem" I got (even though I never played Vasari
)
I started a game on a Large Random 1-Star-Map, so ~40 planets (or 50?) vs 3x unfair AI: TEC, Vas, Adv. Locked teams (so everyone on his own), no pirates. settings all at slow. But map seems quite small, so travel times between stars are real quick.
Here's the current situation:
All of "us" have the maximum fleet size and probably filled with ships
I am the "economic" leader: I hold the biggest share of the System and have the biggest income in credits and ressources.
I grabbed all but one planet from TEC, but there he has his full fleet gathered
I have currently 4 ways of expanding:
1 Phaselane goes to the Vasari planet where he has (I think) his full fleet gathered.
1 Phaselane goes to the TEC planet at the other side of the galaxy where he has (I think) his only planet - with his full fleet
1 Phaselane goes to a Shipyard where a massive (maybe also a full fleet) fleet of the Advent AI is. ( I had no battles with this AI)
1 Phaselane goes to the Sun, from where I can either get to the Vasari planet (see above) or the Shipyard with the Advent guy.
I have no ways of scouting the area behind because non of my scouts gets through those fleets.
My fleet is somewhat spilt: 33% is at the planet beside the TEC-hideout (and near the Shipyard)
66% is at a planet near the Vasari.
The Vasari is quite agressive and constantly launches heavy fleets at me.
So my logical course of action: to send my whole fleet at the TEC and take him out is a no go. I estimate I'd lose at least 2 planets - but probably 3-4 - to the Vasari in that time.
Once when the Vasari had his fleet sent to me I kept a small defending fleet and at the same time sent the most of my ships at his planet. I took out is RA-portal and some defenses, but before I could deal severe damage he had his fleet back there. I then tried to take out his lvl 10 Cap ship while sending in a constant stream of replaced RA-ships. But at the time I noticed that his ship gets repaired faster than I can inflict damage (duh!) I had lost 2 of my cap ships and his other 8 caps there had gained quite some experience.
So basically I cannot beat him at his planet - but he can beat me neither at my planets.
TEC: could probably beat him but only if I could devote my full fleet to it - which I just can't. TEC has been quietly sitting at his planet for a while now, but not sure if TEC really wouldnt try to attack me anymore 
Advent: it's neutral territory and as I had no battles with them before I just don't know if I could beat them...but also only with my full fleet I guess, which again would open up my planets for Vasari invasion.
Is there an idea on how to break that stalemate?
I was thinking on opening up teams, but I dont think this would really help me. Because I couldnt really fullfill any "destroy structure" missions I think.