Many patched tires on cars over the last nearly half century. All good no problems.
PLUGS.. not so good, they do damage the tire and there are other issues.. I might plug a tire if I have no other choice (I carry a plug kit) but Patching is way way way better.. (I also carry a patch kit but dismounting the tire from the rim, not so easy).
True story: Two RVers (One me) purchased the same TPMS at about the same time, Mine arrived and I installed it.. The other RVer received his at a CG and tossed it in the basement intending to install it when he got home (I intented to install mine at home too, but then that's where I was when it arrived).
Well, both of us hooked up (4 down) and hit the road.. About 2/3 the way to my destination BEEP BEEP the dang TMPS starts alarming, Left front tire on towed alarm, Sure enough it was low.. Fired up ye old portable air compressor (I carry like 3 of 'em two 12 volt (Which reminds me I need to move one to the car) and re-pressurized.. Over the weekend I added air every time I went somewhere since it needed it.
Finally on Monday I added air, drove to the tire store and found that just like you.. i'd been Screwed.. The store removed the screw (Road hazard warrnaty) and patched it with a proper internal patch.
That tire was on the car when I junked it years later.
The other RVER, 15 miles from the CG people are honking an pointing.
The left front tire on his towed had gone flat, shreadded, and done over 2,000 in damage to the wheel well and fender around it.
Question: How much did my TPMS cost me?