A Pioneer BH270 travel trailer has a GVWR of 7,700 lbs. That's the max weight it's spec'd for, fully loaded, and that weight should never be exceeded.
Short of getting actual weights, you can estimate that 10-15% of the GVWR rests on the hitch and the remainder on the axles. So about 770-1105 lbs on the hitch and 6595 - 6930 lbs on the axles. A pair of 3500 lb axles should support 7,000 lbs. So there's not much margin, but your max allowable weight is less than the axle rating.
One potential problem is if you're not towing level. Towing nose high or nose low can shift the weight between axles, overloading one and perhaps exceeding the spec. But it seems more likely that you got an axle with weak metal. The dealer said no, but you should discuss with the axle manufacturer directly. They're the ones who say yes or no to the dealer. Depending on what the dealer showed them in the way of pictures or how they described the problem, could have influenced the answer.
Call the axle customer service directly. Have pictures ready that show each axle, so that the problem is clearly seen. Have pictures of the labels so they know exactly which axle you have and can probably trace back to a production run. If it's a metal problem, you're probably not the first to have a problem and they may look at the label and recognize the problem.