So today I applied system updates as usual and also went through my package list and compared it to my Fedora system. I uninstalled some packages that sounded useless / I don’t need anymore to keep my system clean.
After this (and a reboot), THE PROBLEMS ARE GONE!
So I tried to replicate what I did on my second eos test install. I updated the packages one by one. No update made a difference. Then I went through the packages again and removed them one by one.
Turned out:
After removing tlp the lags are gone. No problems anymore with connecting the controller via bluetooth!
Now I know why this worked on Fedora. It does not come with tlp preinstalled, but both EOS and Manjaro do.
Btw., just as a side note, I usually had lagging problems also on the desktop. Lately it felt like KDE Plasma was dropping frames every now and than. I just got used to it because I don’t need the smoothest desktop, so I ignored it.
After uninstalling tlp, the desktop and animations in general also feel a little smoother 
Also this is not the first time I had problems with tlp. Already had them on my old machine. It caused lags and wifi problems.
Maybe this buggy thing shouldn’t be preinstalled by default. It is nice to save a little battery, but this is not needed on non-mobile devices and only causes problems, so users should be aware of it. Having this preinstalled causes users to experience some weirdness in some cases without knowing why. Possibly make it clear in the installer that tlp will be installed. Could be a checkbox in the what packages to install page in Calamares (where one can select which desktop to install).