A used Samsung galaxy phone?
Don’t know if the earlier models offered this but my s5 has a battery that can be easily swapped out and has ant+ compatibility (should you later want to add ant+ speed/ cadence/ heart rate/ power metering to your ensemble)…well, you never know!
And Bluetooth smart compatible to access those functions via a slightly different 2.4ghz protocol.
Don’t know how you’d mount it to your bike.
Me, ive got an sp gadgets phone mount from eBay for £20.
The glass screen protectors can be had for a couple of quid.
I’d recommend some kind of protective case with a lanyard.
First rule of sods law…anything not secured to your handlebars with a lanyard will jettison off at the earliest opportunity.
Slightly above your budget..maybe nearer £130 all in?