I found an article that seems to explain how to set everything up [1]. Sorry for medium, should work in incognito.
Interestingly, I almost bought standalone KVM switch when I discovered that my monitor already has one. Three years after I bought the monitor, I finally understood why there was that fancy blue USB 3 B cable in the box.
Also Logitech MX keys keyboard and MX master 2S mouse support three bluetooth/logitech adapter devices. So I can use this setup occasionally with a tablet and with my old mac that has hardly functioning keyboard and trackpad.
Logitech and Dell devices are worth the money.
[0] https://www.dell.com/si/business/p/dell-u3417w-monitor/pd
[1] https://medium.com/@ningzh/share-a-dell-u3417w-monitor-betwe...
There are BT keyboards and mice that support multiple connected devices and let you switch between them. My keyboard and my mouse can support three each, and I use the functionality daily (on Mac, iOS, and Win10 devices).
For the combo I've got, at least, it's rock solid. My keyboard can also plug in, so at the inconvenience of needing to fumble around on the side of it to toggle the "BT or wired" switch, I could support four devices with it (I don't, but I could). All without any external hardware (KVMs, say) or extra software.
[EDIT] to give an idea of the ease, changing the input on my monitor is the most annoying part of switching devices (for the two that are connected to it) since I have to stretch forward to reach that button, and probably takes about 50% or more of the total time of switching. Mouse and keyboard each take maybe 2 seconds, total, including BT syncing to the other device. Since not all of that's hands-on time so you can be initiating a switch on the second one while the first is syncing, I bet I usually switch my inputs in about 3 seconds, total. Monitor's probably another 3-4 seconds on top of that.
The one exception is when I also have BT headphones connected to the MBP and the keyboard idles into disconnecting - for some reason connecting the BT headphones when the keyboard and mouse are already connected is no problem, but connecting the keyboard when the mouse and headphones are already connected almost never works.
I've got a set of Jabra Bluetooth headphones, and I don't see the problem you do with disconnecting my Keychron, but they do like to connect to all my devices at once, but then are only capable of actually listening to one at a time and tend to get "stuck" on whatever the latest thing was to make a noise. Pair that with the fact that notifications exist and often make noise, and I not-infrequently have it switch away from music or YouTube or a video call or whatever to some other device that just made a notification sound, then get stuck on that device.
If you haven't, you might look into power savings settings for your keyboard. You can change some of that stuff with key combos. I'm sure it's documented on the Keychron site, but a lot of it was printed on a card that came with it, too. I had some powersavings-related annoyances that I solved by basically shutting off sleep mode. Battery life's still acceptable, so no big deal (for me—some people might need weeks and weeks on a charge).
Complaints: it's easy to accidentally hit the light-pattern-cycle button on the top-right, or for a cat to do the same, and I always want it on "steady light, no pattern" (I hate flashing lights on keyboards) but then have to hammer the button twenty times to get back to my preferred setting if it gets pressed. This happens maybe once a week, and is annoying. I also had problems with it dropping into sleep mode a lot when I didn't want it to, but IIRC there was some settings-modifying key-combo in the manual that mostly fixed that (at the cost of worse battery life), and keeping the backlight on its lowest light level (rather than off) during use made the problem 100% go away.
https://www.keychron.com/products/keychron-k8-tenkeyless-wir...
$69
My current mouse: Logitech M720 Triathlon. Basically a boring, normal, easy-to-find lowish-mid-range BT mouse. One of the thumb-buttons on the side handles device switching. Tap once, the light under the current device (1, 2, or 3) comes on on the mouse (those lights are off normally). More taps and it starts cycling through them. Four taps and you're back where you started. Simple, quick, and reliable. Works fine. Has a weighted free-spinning scrollwheel (though you can push in a switch to make it feel like a more traditional wheel) that took some getting used to (that is: I hated it at first) but I really like it now.
https://www.logitech.com/en-us/products/mice/m720-triathlon....
$40 on Amazon
In the past, I've also used a Logitech K380 BT keyboard, which was (and looks like still is?) TheWireCutter's recommended BT keyboard. It's very much a travel-type keyboard, though, so I don't love using it at my desk. It does support three devices at a time, though, and that functionality worked just fine while I was using it. I keep it around just-in-case, since it's tiny anyway, but don't use it much anymore. Nothing wrong with it, though.
https://www.logitech.com/en-us/products/keyboards/k380-multi...
$30 on Amazon
(no affiliation with, nor particular loyalty to, Keychron or Logitech)
(FWIW, I use Sidecar every day and it works really well.)
Sidecar works great for me, too. I just have no occasion to use it where I’m not on VPN.
This is great if you want to use the iPad app version of something rather that just needing an extra screen (Sidecar).
Note that it doesn't like when any of the computers involved go to sleep. There's a reason you can bind a hotkey to restart the server.
They haven't released this technology yet, so we can't test it out. Just an FYI to anyone who looks to try it out.
As an aside, I have a Mac Mini beside a MBP in my normal setup. My BT keyboard and mouse are connected to the mini. I wish I could do this as easily to jump between machines.