Meow Ishmael
The above is pretty good advice, although I wouldn't necessarily recommend tractoring Hornets that are armed with phGs - tractor range is ~2.5 which makes phGs pretty nasty.
What I would recommend is getting one of the F versions of the BC series (one with 2 PFs/INT). Fighters will always concentrate on other fighters/PFs first. Launch your PFs and follow them in. The fighters will shoot your PFs (if they are INT they will probably die very easily, but should have served their purpose) allowing you to close on them and wipe as many full groups out as you possibly can, then open the range to recharge your weapons, hopefully having taken at least 2 groups out (btw I would save suicide shuttles for the carrier itself). If any of your PFs survive this, recall them/it and use it again on your next run.
Alternatively, instead of hitting the fighters on the first attack run, hit his ship with absolutely everything [full alpha + 2 suicide shuttles when on top of him] (except the 2 FA dissies which should be offline), use H&R on his shuttlebay, open the range and recharge. This should leave his ship gutted for power with a dead shuttlebay (ie cannot land or launch fighters till it's repaired). Now he lacks the power to escape and you can pick him off. Be mindful of being tractored by him if you overrun him (to prevent the range 0 ESG hit).
Glad you're trying Lyran out. I love flying them.
One more thing. Toten suggested a HET. Lyran BCs do NOT have a 100% HET. Try it and you can guarantee you will breakdown. Also, you can avoid having to turn down the ESG slider and phaser slider if you don't have weapons set to a higher priority than movement. Only thing I do on the power management settings is set ECM to 4, the rest I leave at 5, unless in a knife-fight, and then I will set weapons to 4 also. Also, on the ships with 4 ESGs, set the slider to 50% from the start. You'll then be able to charge phasers, then ESGs then your central dissies while maintaining high speed and ECM. I treat Lyrans as ~3 turn loading ships - yet again it comes down to patience