The easy way is to have one of the expansion packs that allow you to have a vendor in Magincia. (rustic, gothic?)
Set up a pet vendor, transfer all the pets to the vendor, and price at some amount that people would not buy.
Transfer skills over from one character to the other. Make sure you're properly scrolled for the transfer
Then, go back to the vendor on the new character, and reclaim the pets off the vendor into your stable.
You'll have to rebond them all, via potion or the normal way, before using them.
note that you don't HAVE to transfer all the pets, just the ones you plan to use. Though, you can't really access the ones in your stable that you leave behind, without getting skill back.