Read the Max Load lbs/kg embossed right on the tire sidewall. You need enough load carrying capacity for the axle rating, which is probably 5500 or 6000 lbs (each) on that trailer. Verify that axle load rating, but a 5W must have axles large enough to carry at least 80% of the GVWR and tires to match. 80% = 10,240
I'll guess that an ST227/75R15 in LRD is about 2540 bs (it varies a bit by make and model of tire), so barely adequate for a 10,240 lb axle load. So yes, LRE tires would be the better choice by far. That would gain you approximately 300 lbs/tire in load capacity. Alternatively, use a slightly wider 15" tire - most trailers have sufficient wheel well space for a wider tire. St tires wider than 225 may be hard to find in the 15" wheel size, though.
It appears that Trailer King (available at Walmarts) has a LRE 10-ply in 225/75R15
https://www.walmart.com/ip/ST225-75R15-10-Ply-Trailer-King-II-ST-Radial-Tire/54732677
For a better grade of tire, check out the Maxxis STM8008, which includes an ST225/75R15 LRE tire.
http://shop.maxxis.com/c/m8008-st-radial-tire?action_type=switch_product&selected_cat_keys=82035.1071148.0.0.0&selected_product=385475406683b91fdce26a848ce96a2e&redirected_post=1