I ended up in a very similar situation a while ago while doing assassination missions in my (unengineered) FDL. Saw a mission for 2 MEF, thought it was too good to pass up so just took it without reading.
Arrived at the system, no target, scanned, over to the planet, USS, dropped in ... Elite Fed Corvette.
It was one of those scenarios where the ship is "attacking" a trade convoy (T6+2 Eagles), so I thought I'd just let him engage them, then the 4 of us could bring him down. Took him all fo 5 seconds to turn to me and just open up. I dashed over to the T6 and maneuvered behind it, hoping I could get a few stray rounds to land on it and get some help, nope, 'Vette just stopped firing when he was in the way and the convoy did nothing. Shields were down to 1 ring, so I took off. One weakness of the 'Vette is its top speed. Got out of range, dumped some SCBs and got back up to 3 rings.
I knew the 'Vette was maneuverable, but only for its size, it still couldn't keep up with an FDL. So I shifted 4 PIPs to SYS, turned and headed straight at him, dodging the PA shots on the way in. Got beneath him, pulled up and got on his tail. He boost-flipped on me and hit a good salvo with the PAs, I popped chaff and got back in behind him before the PAs could fire again. This started a back and forth of me getting behind him, him boosting/flipping around and opening up on me, then me returning to his blind spot. After 3-4 passes, I realised I was losing and I needed to ditch my SYS entirely and shift to ENG to stay in his blindspot more. I got his shields down a bit before he took mine out; targeted his powerplant and just kept staying above or below him, feeding round after round into his PP.
Destroyed him with 6% left on my hull. Was a great fight.
You can't go toe-to-toe with a 'Vette in an FDL, you have to out-maneuver them. Boost into their blind spots (behind, above, below) and focus more on staying there than dealing damage. Remember to use your vertical thrusters to help get you in their wake during a boost, just boosting straight at their tail in an FDL will almost always overshoot and you'll get punished. PIPs management is important, I shifted to shields when I knew heavy damage was coming, but my biweaves still couldn't take that much punishment, and I ended up going 4 to ENG for most of the fight. You take more damage if you get hit, but the extra maneuverability means you get hit less often. Save chaff for when you mess up and you're going to take heavy damage, this 'Vette was using 2 huge PAs (IIRC), so the chaff wasn't that useful, but still better than nothing. Their shields can take some time to get through, but once they're down, you can start targeting subsystems. In retrospect I should have gone after his drives, not PP, I would have disabled him much faster that way. MCs have terrible breach and I ended up chewing 90% of his hull before his PP gave out.
A difficult fight, but certainly winnable. Lots of fun either way.