Hi,
From the ribs and flesh, this fig could be a smyrna type or "black mission" for me. This is not a BT, as ripe BT is brown ... But just that one fruit could be off color.
It seems to me that the owners have already trimmed that tree ... for the lawn mower driving but next harvest might be more tricky for them.
The problem with such trees is that they make the ground dirty from the falling spoiled fruits and attract yellow jackets and a lot of things that you would like to keep outside of your property ...
In your zone Aaron4usa, you could have started with cuttings in July. I'm in Zone7 and I do and always did. Just ask for one and 2 years old branches.
On such mature and slow growing trees, a branch of 30 or 40 centimeters is likely to have them both. Taking 2 or 3 should be enough to get a new tree started.
On such encounters, I always try to pack it all at first time . Most of the times when asking to get back, it will not happen ... as some people even happen to kick the bucket in between ... Ask or read the post of Snaglpus about getting back :) .