If you were told that it's relatively uncommon for this particular transformer to fail, the slightly lower copper losses of larger secondary wire most likely won't save you enough heat gain to make any difference. That is, if it would even fit in the existing winding window.

The trafo probably has something like M19 laminations, which has significant core loss and runs hot, even if moderately loaded. It might be best to send it to Heyboer, and have them build you a similar trafo with M6 lams to ensure long-term reliability.

Thermionic