I would upgrade to radial tires and I would upgrade to Maxxis brand. And I would try to go up in size to get more fudge factor in the weight capacity of the tires. I've had excellent results with Maxxis trailer tires on two of my 4 trailers.
M8008 ST Radial
Note thay have size ST204/75R14C (6-ply rating) with 1760 pounds weight capacity, as well as size ST215/75R14C with 1870 pounds weight capacity. Feel around inside your wheelwells and see if you have enough room above and on each side of the tire to go up about 0.3 inches closer to the top of the wheelwell and about 0.25 inches closer to each side of the wheelwell. If you have that much clearance, then go for size ST215/75R14. If not, then you'll probably be happy with size ST205/75R14.
Maxxis tires are sold by DiscountTire in this neck of the woods. They're not on their website, but they can get them for you.
I wouldn't buy those Chinese bombs your dealer is trying to pedal. If you want to buy tires from that dealer and he doesn't carry the Maxxis brand, then go for Goodyear Marathon in size ST215/75R14. Granted, either the Maxxis or the Goodyear will cost you a lot more than the Chinese bombs. But you get what you pay for in trailer tires. And I hate trailer tire trouble.