I'm currently debating restarting my game.
By the hints that you guys gave me, I managed to correctly ID which systems were held by the Precursors and which by the Wormhole aliens. After defeating two Precursor fleets (sadly, just small fleets with a single sensor station world, no ruins or other neat tech to explore), I had a wormhole appear at Barnards Star - the gateway system which I was using to explore the galaxy (since it was the only path out of Sol that wasn't blocked by massive Wormhole fleets). I moved my entire fleet to the wormhole, and over the next couple of months, I defeated three different Wormhole fleets, including one which had a 64,000 ton ship, with a shield strength of 1,500.
However, with each of these battles, I lost 20-30 fighters, usually all of my carriers, and usually one or two of my battlecruisers. I literally cannot replace my ships faster than the wormhole spawns new ones.
And just now, one more ship has spawned from the wormhole - another 64,000 ton ship. After losing the 10 remaining fighters, and losing one BC per volley from the enemy fleet, I decided to check out SM mode to see if I could win this battle.... The answer is: no. Even though I was inflicting around 700 damage per volley of my own, the reload time of my weapons, without a fighter swarm to keep their shields hurting and to absorb damage, their insane shields are recharging to 0% damage between each time I hit them.
My fleet cannot flee to gate out before they will all be killed. The entire upper echelon of my commander corps will die, and although I have even larger, better ships being built, I won't have another fleet for something like 5+ years.
Given that I've seen ~12 ships spawn from this thing in less than 6 months, I don't see how I'll ever be able to defeat 5 years of buildup.