I ended up ordering a new FT-891. Hopefully it'll be here tomorrow and I'll be OTA tomorrow night. It had received pretty good reviews everywhere, although to be fair so did the FT-897 and 991, and the IC-7100 which I was also trying to hunt down. The clincher ended up being the availability of a factory warranty, along with I was a little optimistic about my odds of finding an FT-991 within my budget.
So my plan going forward is to have everything dialed in by Field Day.
I had tried all of that stuff with the TS-140, testing the transmit in CW and AM and not just SSB, using the TX button on the panel and not just the mic, etc. There's just no RF coming out of it. For the moment it goes on to my "projects when I have the time" list, but in a month or so I'll likely just list it with a "doesn't transmit, only good for RX and parts" disclaimer.