Kyle's pretty good in the La-7..but I met him in quite a few engagements, and shot him down a few times in a 1-vs-1 situation in the La-5FN. The only instance where he shot me down was when it was a vulch-situation(where he usually enters the scene). The rest cases I've been shot down were all from his N1K2s.
He's not as sharp in the La-7 as he is in the N1K2(and also, not as sharp when flying alone, without the usual cronies he brings along. In fact, his La-7 is pretty much basic - disciplined, but nothing extraordinary. The only exceptional thing I notice is the knack for long-range shots, hitting 600~700 yards with even the ShVAKs(or maybe Beresins). You think you gained enough range to extend and then wham, pow, a burst of 20mms and you are damaged or downed.
Therefore, I'm pretty sure you missed something - maybe he had an initial E advantage you didn't notice, or maybe you misjudged something. Whatever it was, it was probably something you didn't catch. My bet is on in truth, he came down on you with some sort of E advantage.
So, I bet it went like this; you decide to extend away, he targets you with an E-advabtage you didn't realize. He catches up with you briefly before you reach full speed on WEP(by this time, you two are probably down at deck and with only each other).
Then, he probably bursts a stream of long-range shot which a few pings land on you, and in disbelief, you decide to jink. That small jink was enough E-loss and he probably got to catch up with you. You try a few maneuvers, each time trying to get a situation where you can out accelerate him.. and every time when you seem to be gaining distance he'll do a pretty accurate spray, and you have to turn again.
It's a pretty typical behavior in notorious N1K2 drivers. They shove 20mms up your arse at ranges you can't believe, whether you are jinking or not.