Just tried PS4 remote play, seemed terrible at first.. Though that was because I didn't know you had to enable the high framerate option yourself, looks great now.
Though it just reminded me how badly I want a Vita with L2/R2 (´・_・`)
I remember there being a Vita case that added L2/R2 for remote play.. Was that any good?
Go into Power Save Settings in Settings on the Vita and untick "Use Wi-Fi in Power Save Mode".