Can a LA add a reg requiring cars working via a base carry signs for that base?

Unless challenged in court, yes.

If doing private hire work it's better for the vechicle to be identifiable to help make contact with your customer even for Hackneys otherwise if a unmarked Hackney turns up the customer will have to approach it to find out if it's there cab and even if it isn't there's a fair chance they will end up in it then you turn up two minutes later for a no job.