Well I don't think you will ever be able to get a full length into the space with enough bearing on both plates. So looks to be leaving you with option 2. I'd be looking to strip the bottom row of felt and replace.
Cavity has its benefits. Very mainstream when it comes to finance, insurance and local skills. It's not always the best solution, up your way TF is more common.
I had a automated gate about 1.5m high and put it inline with the boundary about 300mm back. I pressed the button as approaching or just pulled up at side of rd. It was a decent width rd though. I was using an existing access which already had gates on it though. Never heard anything from council.
Makes me wonder whether "installing" some scrap gates across an existing access before applying for permission is a good "hack" bet you can find scrap gates on marketplace
I also ran the numbers on single glazing even though you wouldn't be allowed it in UK regs. Interesting how big a difference in u value between single glaze and 2g vs not so big a difference between 2 and 3g (if the numbers are accurate) 1/2/3 5/1.2/.7
I might approach my next place differently instead of fabric first ill do economic fabric first.