If this was caught when another party tried to claim the child, it might be better to have them claim the one you client didn't claim. Sounds like they are doing some switching back and forth, and if they have to wait for your client's amended return to process to free up the dependent, it might be a while.

The above comment is dependent on "there being no change to either return regardless of which dependent child is claimed."

Some times it makes sense to do things the expedient way, so long as you are not changing the tax amount of either party.