Hot glue: you've taken a page from the Ghurd Book of Electronics Repair.
If there's a voltage across the fan's leads, when the inverter is loaded and hot, but the fan isn't turning, then it's probably the fan's fault. You don't have much to lose if you just cut off the leads of the old fan, and splice in the leads to the new fan. Mounting the fan in the old fan's spot requires matching holes, space in the case, etc., but maybe the hot glue on the outside approach is fine, too.
If no fan will work, because the leads from the board aren't powered at all, then a different approach is needed. You could get a snap-disk thermostat that trips ON above 50C, roughly, and off below 40C, and attach it to the inverter's outer case, or some other convenient spot that will get hot under high load. Use heat-sink thermal grease, not hot glue. This thermostat, in series with the hard-wired fan's power supply, will prevent it from running when the inverter isn't hot. I don't have a thermostat catalog in front of me, so I don't know if that specific temperature range is available, but something close will do.