Wireless devices can be problematic. Maybe you should buy from a
local store, especially if range is your concern. Do not believe
Logitech's (and probably others) claims about transmission range.
I have not used Bluetooth. My Logitech radio frequency wireless
devices have worked very well, but I don't care much about range.
Good luck.
Both have their drawbacks IMO making one no better than the
other. Go for one that gives the best battery life and has a
mouse with a docking cradle for recharging. Try to get them with
rechargable batteries built in.
Rechargeable batteries, including built in, would be a waste on my
cordless keyboard since alkalines last for at least six months.