"purchased a passenger car tire to replace a flat"
One tire only? If you don't replace both sides of the axle at the same time, there will be problems regardless of the tire types. Then, since it was a newly mounted tire, there's no way to know if the rim was damaged, the mounting was improper (tire cord torn during mounting), the air pressure wrong, the torque on the lug nuts wrong....
So at least five things may have been wrong enough to cause that failure. Can't say that the sixth (passenger tire) alone was the problem.
Michelin and Goodyear make radial trailer tires, in some sizes at least they only make radial trailer tires. If I had to choose between no-name Chinese bias play trailer tires, and top brand reliable trailer tires that happened to be radial...I'd trust the top brands. And yes, find the extra dollars to pay for them, one way or another. I'd rather take a Nantucket Sleigh Ride than ride on Chinese tires.