It's anyone's guess why it squeaks, but as you heavily use it, it may be wearing the buttons. I didn't see where it doesn't operate as original, only annoying squeak and white timeout, correct?

If you want to stick with the same form factor/color display/rechargeable, look at the MX880. It will require a complete new program but it's essentially the same look/feel as the MX810. Personally I would spend a little more coin and get the MX980. Different type of buttons, and very customizable.

Pretty much all current URC remotes will work with your MRF350.