What PSP firmware do I need to run both new UMD games and homebrew?

I'm currently running 3.71 M33-4 on my old PSP fat that my son mostly uses and now that I bought him a few newer games they would not start and said that the firmware must be at least 5.55. But if I upgrade, I won't be able to run my homebrew emulator softwares anymore. Is there a way I can have the best of both worlds? What firmware should I upgrade to?