Are none of the available programs drop-offs?

Here, our Y does a homeschool gym/swim for kids from 5-18. It's a drop-off program, once a week for about 3 hours. Our DS8 has been doing it since he was 6. He gets swimming instruction, plays the kind of games he'd play on the playground (freeze tag, red rover, etc.) and also plays in organized sports like soccer, basketball, badminton, etc.

The Y also has drop-off classes for all sorts of other things, including art, but most of these are not specifically for homeschoolers and take place in the evenings. I do know of many drop-off classes for homeschoolers during the day, though. Call the art centers in your area. We have a lot of them in our local metro area, and about half of them do daytime classes for homeschoolers; many of them started offering these when approached by a homeschooling mom who said, "If I can get at least 4 kids for a daytime weekday class, what can you offer me?"

I am also interested in PT enrollment, but for academic classes like lab sciences and foreign languages - things I have trouble accommodating at home. Like you, our public schools don't offer PT enrollment. I haven't looked into the private schools yet.