From your post, I presume you don’t know where he lives. Could you find out given that your fathers know each other?
If you can, send the invoice there. Alternatively, if you can get his email, you can email the invoice to him. That should be good enough. I hate to suggest it but is he likely to be on Facebook? If so, you might be able to send him the invoice that way.
You arguably don’t even need the invoice unless it was agreed that he would only pay on receipt of an invoice. If the agreement was, you’d do the paintings, he’d pay $200 for each and you’ve done the paintings, that should do. Now that you’ve emailed his girlfriend that you want an address to invoice him, you might be stuck with having to invoice though.
I presume the reality is that if he doesn’t pay, you are not going to sue in any event but if you had to an emailed invoice should suffice.
You will probably need to know where he lives in order to sue, if you were going to (at least down to the county level). If you do have to sue this might help?
If the arrangement was with him personally rather than the business, I’d issue the invoice in his name - you can still send it to the business address if you have to, just make sure the invoice is in his name rather than the business name.
Alternatively to all of the above, you could just assume that he knows he’s got the paintings and should therefore pay you. He knows how to get in touch with you so he is deliberately screwing you over. Is sending him an invoice going to get you any money? Are you going to sue him if he doesn’t pay?
If not, you might as well just write him off as an asshole. Would your father be able to get in touch with his father to put some pressure on the deadbeat acquaintance?
Welcome to the joys of business life. Your friends and acquaintances will often screw you over worse than any stranger.