Easiest way - take the two shots. One with everyone holding with their card loose and relaxed, one posed as desired. Make it fun, use it to help people relax, get some smiles started. Let them play around with concept - as long as you can see name on card. I use approach on occasion and no one has an issue.
A ring booklet booklet of recipe cards works great. Small, easy to pack and just flip to new card. It also keeps the cards in order so worst case know that name 6 belongs to 6th person shot. Maybe do something in post with (some of) them afterwards for a chuckle if think audience open to it.
Any other method I can think of risks getting names mixed up and that would be far more embarrassing. Face holding name is foolproof.