True story: A fellow RVer reports purchasing a Pressure Pro TPMS, well it arrived while he was camping so he tossed it in the basement storage intending to put it on when he got home.. 15 miles from the CG, people are honking and pointing, and he found the tire on the towed was totaled, As was the rim, the brakes and the wheel well.
I put mine on when it arrived, 2/3 the way to where I was going and BEEP BEEP. So I added air to the tire on the towed (Same corner as the guy above) and continued to add air all weekend, Monday I added air and headed to the tire store.. They investigated and found that like the O/P, I'd been SCREWED, they removed the screw, put in a proper patch, re-mounted and balanced, and no charge (Road Hazard Warranty). Tire lasted several years.
I would have no problem with a patched tire, Never have had a problem with a patched tire, Used to work in a business where we had many patched tires (Nails, not screws but same effect).