I went to the local parts store and they did not have either the triac
or the diac. They did have a TNE5646 "Triac with Internal Trrigger
Diac", which I purchased. This is also a 600V device and it handles
10A.

Can I simply install this in place of the BT137F and jumper between
the two holes on the circuit board for the missing diac?


You should be able to, give it a shot, worst case it won't work.