It would be extremely difficult to beat their fleet with with what you have, completely impossible if it's one big doomstack, so avoid their fleet at all costs. What I've done in my playthrough is I got their doomstack to chase after my transport ships (the AI seems to take priority in eliminating them). I would have their fleet warp after me, go orbit the star, then move away and leave their fleet in the middle of the solar system and warp away, forcing them to go to the edge of the system again and giving me time before their jump drives can catch up. They have tier 4 thrusters, you should have them as well otherwise this won't work.
In the meantime, fight their allies, they can be beaten far more easily and occupying their planets will provide you with a lot of warscore. If you feel confident enough, you can even capture the planets of the AE itself, as long as their fleet is distracted you can attack their planets facing little resistance. Don't go after their capital, it is heavily fortified by fortresses and if you take too long to destoy them the AE fleet will have the time to engage and anihilate you.
Keep you fleets near their doomstack, but not in the same system unless you absolutely have to (other than the bait-transport fleet). If they send reinforcements (a small fleet roughly 10k) or transport ships you can destroy them and research the wreck to get jump drives yourself, that will help with outmaneuvering them.
I've won against them with this method on multiple occasions, but you might need to reload your saved game a copule of times, Timing is absolutely key in making sure their doomstack never fights yours. Last tip, if you click on them it will often say they are following Fleet-xxx . You can split that fleet and leave only one ship in it, they will continue to chase after that one ship and the rest can get away.
