I'm in New Zealand and I use this
https://www.spotlightstores.com/craf...-multicoloured in my Big Shot Foldaway.
I don't know if it's available in the US, but the key thing is that it's Cuttlebug C plate thickness. So if your existing mat is C plate thickness then it's worth trying in your Big Shot. The sandwich I use is basic platform, magnetic C plate, die and cardstock, one cutting plate. Or if you still have all your cuttlebug plates, use the A plate in place of the standard platform. Just - as you said - don't force it, and start by experimenting with a small die.
HTH