Sorry, no suggestions for you, Saro, as I have a Vita and a PS TV for Remote Play.
Regarding the 60 FPS option: when Sony introduced that feature, they must have completeley reworked the protocol. Before, there was annoying lag in the best of cases. And every time there was a hiccup in the connection, the audio got out of sync with the video. That issue didn't resolve itself over time, forcing you to restart the connection if it became unbearable. I was on the brink of dismissing the Remote Play feature completely as nice but gimmicky and ultimately useless.
After the patch, I Remote Play all the time. It works much more smooth, even on 30 FPS games. The lag is reduced to a point where you can play fast action games without massive problems. And the audio never gets out of sync now, even if there is a hiccup in the stream, which you can't completely prevent over WiFi.
tl;dr: Remote Play kinda sucked before Firmware 2.5, now it is absolutely amazing. They should promote it more.