I actually found a way to take care of this myself. I went to the Arch wiki and...
I tip my hat to you

, Arch Wiki can be an invaluable tool, and a treasure trove of information as long as you don't have to click all the links, or else you might do a Jumanji and not come out for 20 years.
Plus of course, adapting some of the commands to suit the non-Arch environment.
I am by no means a Bluetooth expert, so I can only guess there, but my reasons for the questions on DE and kernel were two-fold:
Over the last 3 years or so, it has become the practice to place more and more drivers within the code of the Linux kernel, and less as individual packages to be added from Repositories or downloaded from other sources, so newer kernels can provide better access to newer equipment.
At the same time, support for some devices may be dropped if it is perceived that there is not the demand for justifying their inclusion. So knowing your kernel version could help a helper advise whether you could try an older or newer kernel.
The question on DEs was simply to establish your Environment, so that we could perhaps say "Go to here on your Menu, and to that category, and do this or that".
I run 60 Linux on this rig, which include a number of Mints and a number of Manjaroes, so I boot into the one the Member is using if I have it in stock, for easier guidance.
I am guessing on the Bluetooth-specific side of things, but for ever so many packages, there is .conf or similar config file that can be tweaked, and have differing default setup across Distros. It is possible that a difference exists in that setup between Mint and Manjaro that would explain why one prompted for the code and the other, not.
Someone may swing through here with the answer, or if you find it yourself before us, by all means share it with us.
Do consider swinging over to Member Introductions and tell us a little of the Techforay story, you'll find the Gang to be friendly.
Cheers and enjoy your Linux
Wizard
BTW - Timeshift is available in all Manjaro repositories. If the keyboard is working fine, I would install Timeshift and take a screenshot. I can field questions on Timeshift at my Thread here
https://www.linux.org/threads/timeshift-similar-solutions-safeguard-recover-your-linux.15241/