There is no need to retain the broken PCB.
I am intened to mod mine HSS-0136 too. And my plan is to replace all the buttons with Sanwa's buttons. I would throw away the green PCB which you have broken. And disolder the white ribbon cable from the other PCB (the brown one shown in your photo) and then directly solder the wires (to the button) there.
Before you do that, I think you have to upgrade your desoldering tools. Currently how you do desoldering? I see lots of solder remains at the desoldering points. You are supposed to remove those solder with some tools.
I think you have to at least get somthing like this:
https://www.amazon.com/GBSTORE-Sucking- ... B01K72SBWY
With this tools, you are supposed to heat the solder with the soldering Iron and then suck the melted solder from the PCB with this tool. Repeat until all the solder at the desoldering point is being removed. Then the parts would break loose.
By the way, do you know how to wire the buttons without the green PCB?
If not, I could send you the green PCB of my HSS-0136 if you like.
*Meow* I am as serious as a cat could possible be. *Meow*