The QK 12" platform ($10 at OhMyCrafts and it comes with 1 cutting pad) with some shims or the Sizzix Extended (Essentials) Platform with the Sizzix Extended Cutting Pads would work. I have some of the Bigz XL dies and had to buy the Extended Cutting Pads anyway.
You will need something in addition to the QK platform to send the thin border dies through a CB. I bought it thinking it would be enough and it isn't, so I bought the Sizzix Extended (Essentials) Platform and between all the extended platforms and cutting plates I have now, I can get a good sandwich.
I use this in a BigShot, so let me find everything and see what would work in a CB. The QK platform in narrow, but the Sizzix ExPlatform & Cutting Pads might be too wide.
I got a better result putting the die on something the right size and using the smaller plates as shims. The first few times I shimmed underneath the QK platform with various regular-sized CB & BigKick plates and some cardstock and eventually hit upon a good sandwich. Then I decided to just bite the bullet and buy a Sizzix Extended Platform. I've used the ExPlatform enough now that I feel it was worth buying.
QK Platform @ OMC here
OMC also has a bunch of the QK border dies on clearance for $12.50 on this
Extended Cutting Pads
Buying the Sizzix Extended Platform wasn't a waste of money (and it was on sale a while back). I figured it was worth it since I paid only $12.50 each for QK border dies. Also, now I use the Sizzix Extended Platform & Cutting Pads to send multiple Sizzlits through the BigKick. It saves a bunch of time.
Sizzix has free shipping if over $75.
OMC has 10% off your entire order (including sale & clearance items) going right now: YEAREND10