If you can't make something work from China, you could even just get someone to make you an embossed polycarbonate overlay with self-adhesive that you could attach to a PCB with existing microswitches or metal snap domes. It likely won't cost $2/ea, but the MOQs could be much easier.
Your design is common enough that it might even fall into a manufacturer's samples tooling. You'd likely just have to pay for silkscreens and cutting dies if you can't make their exterior dimensions fit.
If you can fit everything on a 2-sided PCB using SMT, the switches could be the front side and the active circuitry could be on the back.
If pricing forces you offshore, Kingley is one reputable supplier from a past life:
https://www.kingleytech.com/membrane-switch.htmSan Teh is another manufacturer from the past, but they specialize in elastomers, which also may fit your price point. Keeping with the above, 2-sided philosophy, the elastomer keypad could be secured to your PCB with sort-of rubber posts, which would be pulled through holes in the PCB. You then have the option of microswitches or carbon pucks.
http://www.san-teh.com/