Welcome to the Sunline Club, FAB 54. Sorry to hear of your tire problems. I can assure you that tires sizes and ratings in Canada are indeed the same as in the US and probably everywhere else as well. It would be helpful to know what tire size was on the trailer in the first place.
Did you by any chance still have a really old original like C or D78/15??
Unfortunately, it appears that Sears sold you a car tire rather than a trailer tire. The 165/80R15 is a smallish, squishy, low pressure tire not suitable for trailer use. Tirerack.com lists only one brand here:
It doesn't matter what brand you have. All tires are built to the same specs, and within a few minor variations the numbers across this table are most likely your numbers too.
You say you have had ongoing tire problems and have been replacing them. What do you have on the trailer now? I can't tell from you post in Repairs and Maintenance, where you say you have been replacing tires, whether you still even have these Sears tires on the trailer.
In any case you should be using a special trailer or ST tire. Goodyear and Carlisle are big manufacturers, but you can also get ST tires from other vendors like national chain tire shops that carry various house brands as well. Club members run a variety of tires including these and Denman and I have Maxxis, both of which are brand names.
Sunline Fan will likely know what size you should have--I'm guessing ST205/75R15, which is the smallest 15" available today. They should be run at their full cold pressure of 50 psi. Here's the Goodyear example from tirerack, but again any other brand will have virtually identical numbers: