
So there was no loan of bitch agreement lodged with the kennel club?, you simply own a bitch jointly.
If you don't have a joint affix then the pups can't be registered with any affix, if she is still in joint ownership when the pups are born.
You can only use your or the other persons affix if one or other of you solely owns the bitch.
Your affix is for your sole use for litters born to bitches you solely own, it's your affix, and the same for the other owner. You could take out a separate joint affix for litters you breed in joint ownership.
The least expensive option is to transfer her to one name (before the litter is born), use that affix for the litter, but the pups will be listed as bred by that person alone, and then transfer her back to joint ownership after.