Having the same issue, BT cuts off after 5-20 minutes of use -- the BT settings button is gone from the right swipe-in column, the blue on/off switch is missing from the Bluetooth Settings window, and it says "Bluetooth is not turned on" under all the devices.
First started on my Surface Pro 6 same day as y'all (around Aug 5th after the update), was on with MS support all day last Monday (they had me reinstall drivers, then reinstall Windows) -- I thought this fixed it because it was working the next day after being left on all night, but after only 30 minutes of use that morning the BT cut out again (so it seems to only cut out after so many minutes of actual *use*, not just running).
Then I was borrowing my partner's Surface Book 2 all last week, working perfectly fine, then it started acting up the same way yesterday! I ran the troubleshooter on both computer.... same thing happened as OP where it caused BT to stop working, and it gave me Radio Signal error which I fixed following a tutorial on the forums here, but BT would still cut out again after a few minutes.
I sent in a Feedback Hub report on the Pro 6. This is so completely ridiculous to me that Microsoft's FLAGSHIP LAPTOPS are being ruined by this update. Like many of y'all, my workflow relies on BT keyboard and mouse so this has completely upended my productivity (mainly because I've spent hours upon hours over the past week trying to fix the stupid thing!)
Posting here so I keep updated if anyone gets theirs fixed!!