iMac 27" late 2015 has major bluetooth issues after Monterey install

We have two Macs on Monterey, 2015 and 2016, both have operated beautifully with every version of it.

You are obviously having problems, but problems on your Mac does not mean problems on every Mac. Neither does a sample size of n+2

Have you tested your Mac with diagnostics? I'd start there: Use Apple Diagnostics to test your Mac - Apple Support

Pending the results of that, solutions may be offered here by those more familiar with BT and its quirks.

I bought 3 computers with the same exact configuration as the user above. I verified my two kids computers and mine were ordered together and at the same time from the Avalon Apple Store. ( all 3 Say iMac, Retina 5K, 27 inch,Late 2015 Processor: 4 GHZ Quad core Intel I7 Memory: 32 GB 1867 MHZ DDR3 Graphics: AMD Radeon R9 M395 2 GB) 2 of these are used for heavy video editing and have dual thunderbolt Promise Pegasus 8 Bay Raid array's (Total of 4), also bought from the Apple Store at the same time, one of the 3 does not. I have tried everything suggested before and been walked through with Apple twice (I also paid for the APPLE Care package but don't even have an idea of how it works but I got working once on one machine but at first reboot, days later, bluetooth does the same thing, it only shows Magic Keyboard and Magic mouse as "connected wirelessly via bluetooth" but only while plugged into USB cable so were now having to take two of our needed usb ports for mind you, I bought 3 newer keyboards and 3 Magic Mouses)and now using a store bought wired mouse to use each of the computers. It stopped showing incorrect passkey after my most recent update (Monterey 12.7.1) and shows each of the keyboard and the mouse in the Bluetooth display only if plugged into usb. and yes the batteries are fully charged and not burnt out, they work fine when used with other newer and older iMacs in the house and batteries are at 100 percent. Super frustrating, quite aggravating and with all we bought that Christmas my Apple bill when paid was close to $24,000. I do drone editing and occasionally need to unplug either mouse or keyboard to connect to our 2 Mavic 3 Cine versions and I have also been through 4 iPhones in 4 years, and my two kids just as many and purchased so many iPads, iPad Pros and keyboards and Macbook Pros, Airs and chargers and blocks that I am ready to throw in the towel and go to something different over this issue. I have another friend going through same issues as posted and it sounds like he's about done too. SUPER frustrating, aggravating and I don't upset easily. I was an IT director for 8 years with a large telecommunications company and know hardware and software. So when someone says well I have 2 of the almost same exact things and they work beautifully, it sounds like you're telling me and the other customer involved, and I'm sure yours do work properly but I have 3 in my house we were using quite often,one daily, one my Sons, the other my daughters, and now are taking up space on desks, I get agitated. I'm sure there are many, many others just like me that have never posted a thing on here before, and just are doing as I have and patiently waiting for enough phone calls to right the error, the wrong. The error in coding, each one downloaded and same thing wasting an hour being hopeful on three $6000 machines even though was several years ago, you have to give up on the company you once had great faith in. Sorry Apple, Im done

I'm with you Mike, I just read your reply, you sound like you're about where I am at with this. Thank you for starting a decent much needed, and too long awaited and the then the reply you received. I had to respond to that as well.

I have the exact iMac 2015 with similar configuration, i7 + Radeon R9 395X. It had the bluetooth issue since day 1, and I gave up (I tried) and have been using wired keyboard and mouse. I think the weak bluetooth is because of its design.