I've been hunting for tall, white, capped buttons that are the right length to lay almost flush on the frame's border (easy enough to press, but hidden).

As of today these are the buttons I've found. Unfortunately none of them are long enough to get through the frame. I misunderstood the labelling, apparently the marketed button length is measured from the bottom of the base to the very top and not just the length of the cylinder.

I think if I glue the cap onto the 9.5mm button I will get what I'm looking for, or I can purchase buttons marketed as 15mm length since the button base is usually 3.5mm.