I think people cover the twill tape to match the fabric they use to cover the bra because, well, twill tape isnt very pretty ;-) I also think twill tape would dig into your neck or shoulders on its own if you have alot of decoration on your bra.

I generally use grosgrain ribbon, two layers stitched together if the bra has heavy decoration. I cover the ribbon with the same fabric as the rest of the bra, unless I can get a perfect color match.

If your coin bra cover is heavy, then ideally you want a bra base that will stay (mostly) up without straps. If you rely only on the straps to hold it up, the straps may dig painfully due to the weight no matter what they're made of!

I use lots of things for straps. Grosgrain ribbon decorated with trim or covered with fabric that matches the bra. Thick elastic covered with the same fabric that covers the bra or (when the straps will be hidden by a top), bias tape tubes. Satin, velvet, or other decorative ribbon lined with twill tape also works.

Also, back to my first point, if you are big busted or the coins are heavy I strongly suggest avoiding halter style straps. Instead, try straps that criss cross in back. Or over the shoulder straps, but with a horizontal strap connecting the two shoulder straps, located high on the back but still below that sensitive point where the neck and body join (H strap). Or, make a halter style strap, but sized so that it reaches below that sensitive point, and add a single long vertical strap to connect it to the band at the back ( I strap).