Yep. Load rating is not absolute, but it related to the size of the tire - a smaller load range 'E' tire may be rated less than a larger load range 'D'.
What PR are the tires you have now? I know I've seen 14 ply 14.5 tires, but can't remember the load rating.
Triple axle? Seems like it'd have to be. You know what the axles are rated at?
Are you sure it's a properly built trailer? 14.5 dayton rims are what 95% of mobile home axles used, and I've seen a large number of trailer built on those axles.... which means the GVW may or may not be correct. Could be someone swapped new axles in as well.
If it's a triple axle you're probably ok - 3 6k axles is ~18k, plus tongue weight, minus a bit for the uneven axle weight distribution. I personally would prefer to see tires rated a bit over the axle rating, ie 3500 lb tires just cause tires are usually the weakest link, but that may not be possible.