I show up and they cannot fit it into the bay!

So I did just that. I took off the front rims and tires and took them over. One hour later they were done. Worst case, I could have done this one at a time in their parking lot!
So if you've got a six ton jack, jack stands and a pickup, you can do the same. The rim / tire combos are very heavy, so be careful or consider taking the rig elsewhere to have it done. I also found out that the lug wrench I got with my ambo was the WRONG SIZE. So this was an opportunity to get the right one instead of being stuck somewhere.
Yes, it's more work, but knowing there are Costcos everywhere and they have a great tire hazard warranty, it's worth it.
If you're thinking about painting or powder-coating the rims, now's the time to do it. Tell Costco to remove the tires and give you the rims back for a few days and see if they are OK with it. They should be, you've already paid for the tires. In my case, I did it after the tires were mounted.
How did it go for you at Costco tires? Did you go somewhere else?