OK so I'm not going to spend a bunch of time extolling the controller that I'm about to suggest except to say that I was also looking for a replacement controller with gyro and back buttons; I even went so far as to drop the gyro requirement and spent nearly $200 (incl. tax & shipping) on an XBox Elite Series 2, which is a fantastic controller.

And then I found this: GCHT Wireless Pro Controller

4 programmable back buttons, acts as an XBox 360 controller when plugged into USB and as a PlayStation controller when wireless. The added bonus of the touchpad in the center (like a PS gamepad) that can be configured as a mouse trackpad put it over the top and made it my new favorite controller, even moreso than my Steam Controller. Full Steam Input support. And less than $40!

Do yourself a favor and try it. If I have one complaint it's that it's possibly a bit too small for my hands, but with all the other bonuses that's a small (hah) price to pay. I can't recommend it enough.