I'm in DFW trying to decide between some type of St. Aug and some type of Zoysia. Both portions of my lawn are fairly shaded although I plan to have a trimmer come out soon to open up the canopies.

I'm leaning towards Zoysia however the property line with my neighbor (roughly reflected in the second pic) has really nice, strong St. Aug so I'm concerned how Zoysia abutting the St. Aug would fare. And to complicate matters, my neighbor just sold his house with a leaseback until July so I'm not sure if he'll give me the sign off to just replace his little section with Zoysia.

Quite a bit of shade there. Palmetto St Aug is going to be your best bet to battle it, along with keeping those trees properly trimmed. It's the most shade tolerant option available

I would remove that first tree. It's a junk tree anyway. Replace it with a Chinese Pastiche.

If your neighbor has SAG and you trim that second tree properly, you too will eventually have SAG.