I went down the same journey and I ended up with (I need something to connect to multiple devices) a G915 TKL and G603. I paid £180 for both with a student discount - although that is a little out of your budget (£150 for the G915, and £30 for the G603).
I don't recommend Bluetooth for gaming, you would want 2.4GHZ wireless.
Something after searching on the rainforest is the "Mechanical Keyboard Bluetooth / 2.4GHz Wireless USB/Wired Type c 3 mode US Layout 87 Keys Rechargeable 3300 mAh Battery Metal Plate LED Backlight Gaming Keyboard For Typists Gamers - Brown Switch" - reviews look good and its cheap. Pair that with a Logitech G703 and you should be golden - it is taking a punt though, I've not heard/ read anything else about the keyboard.