hi what version of TDV you are playing

It's possible. It is however much more reasonable to deal with if you can get your hands on a flight controller. I have played through the game with both a keyboard and a controller and I can say the experience with a controller is much more realistic.
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That being said, you are correct in that you cannot wack him three times with a cruise missile. Use the missile interceptors to your advantage though, as that will deflect the missiles onto his jet. Other than that, bullets, bullets, bullets. And cruise missiles, just know that they won't be the end all be all. What I would do, is if you're in a safe position to, i.e you hear either Chase or your rio tell you you're clear to go in for the kill, go in and tag him with the lts. You cannot lose your target that way. Then just fire away with your guns until either of them tells your to break off your attack. Then turn around if you're still at least relatively facing him. Fire cruise missiles in the interim as you will already have tagged his fighter. Keep repeating this for each round.