Well, it was time to check the cat's urine again.... we ordered home test strips and collection tubes.
Fortunately, I have a wife who is a 30+ year medical lab tech. All I had to do was convince the cat to pee in the blue pan. That required a bag of sand, a piece of cardboard, and a powered staple gun. Use your imagination.

finally used the blue plastic box with the hydrophobic sand in it. I opened the door to the basement that I had stapled the cardboard over the cat flap (to keep him locked down away from the other litter boxes) so he had his freedom. Then I tilted the box, and used the plastic pipette to collect the sample into the mini-test tube.
Wife did the dipping of the test strip, and the reading of the colors after the elapsed time. I was ready to get her gloves, eye protection, a trashbag for a lab coat... whatever she needed.... But she did it like an experienced pro...
"It's just cat pee" she said with a roll of her eyes, and dipped the test strip.
It came back positive... he piddled. It was urine.
Basically, no change from the test a week ago at the vet. Still some red blood cells in the urine. But he's not straining to pee as much as he has been. He's actually used the box several times through the day. We'll see if that continues.