Looks like your problem is that the air leaks out before the tire has a chance to seal. I had the same problem on a dolly I own. Here is my solution, use at your own risk:
Get some highly flammable substance (I used gasoline)
Lay dolly down.
Pour about an ounce of gas into the tire.
Give it about 10 seconds to evaporate.
Light a piece of paper on fire and throw at tire from a few feet away
When the gasoline ignites and expands, it seals the tire. Ensure you also fill it to the proper pressure when you are done.
Tubeless tires require a certain amount of pressure to keep the beed sealed, I would make sure the bead was “Very” clean and make sure enough pressure is added along with the green slime you should be ok.
using a flammable liquid explosion can be dangerous. what isn’t put in the tire if left exposed and nearby is a fire hazard.
an ounce of gasoline is a large amount.
the liquid may be bad for the tire.
safer is a second or two spray of starting fluid from an aerosol can.
Wouldn’t be good for the plastic hubs, either.
I would look for solid tires so you don’t have that problem any more. That’s what I did for my wheelbarrow.
That was my first choice. None of the hardware stores have wheels with big enough hubs. As I said in the OP, the diameter of the tube upon which the tires rotate appears to be about the same as the bearings that are in such wheels.
As usual, I’ve been too busy to do anything about the wheels. I could use the cinch method, as I was going to; but I don’t want to have to do it again later. I ordered some tubes from amazon (I hope they fit – I haven’t opened the package yet), and went to the hardware store to buy some tire irons. They didn’t have any (they thought I was asking for a lug wrench), and sent me to the Napa store (which I didn’t know was there). They verified I didn’t want a lug wrench, and then said they don’t carry tire irons. So tomorrow I’ll take the wheels down to Les Schwab and have them put the tubes in. They just use screwdrivers, which I could do myself. But I know if I do it, I’ll pinch the tube and poke a hole in it. And I can’t tell you the last time I saw a tire patch kit!