When dealing with plastic in the header material stuff like this happens with high temp tools. However, all you need to do once that piece is removed is to cut your own pins with the largest component spare leads you can find. I also got lucky that the PCB had soldering pads on both sides.
First attempt I had 10 of the 16 pins working. Two more rounds of small changes and all but two going now in these pics. Experience wins again.
I should have just gone out to get longer screws than came with the kit to accommodate the slightly higher depth of the module with the 16 count pin header flipped around with the short pins going into the main PCB and the longer ones pointed towards the LCD. Apparently that isn’t what was intended, it was meant to have the shorter pins facings the LCD PCB instead and longer ones going into the primary PCB. Be careful with this part of the build.