Lavochkin fell out of favor with Stalin 42-43, so his LA-7 plans were put on hold except for doing in the field modifications to LA-5's which were called LA-5xx. The radiator was still under the engine not under the pilot, and they carried 2 cannon. Eventually Lavochkin came back in favor with Stalin and was granted a factory and a mandate to build "The Boss" a world beater fighter plane in 44. Stalin was worried about Allied aircraft developments in the West(Always the forward thinking Paranoid). This became the LA-7 we know in AH with the radiator moved back under the pilot, 3 improved cannons, a bigger more powerful engine than the LA-5 , and improved external streamlining.
The LA-5FN can be concidered an early generation LA-7. They were the same plane. Politics and orders not to produce anything but the LA-5 kept Lavochkin from calling it the LA-7.
-bustr-