"They're having a great time"

The accord with the verb is still as if it's a plural pronoun. Just like with "you" which technically also used to be plural, doesn't take the old singular second person forms: "you have" not "you hast"

Don’t force the verb form.

“I’m worried about my kid at summer camp”

“Don’t worry, I’m sure they are having a great time”

Sounds very natural to my west coast ears.

I believe it's typically only natural when the name/gender hasn't been established.