Ah thank you! That has indeed solved the issue! Solution was in this post: Bluetooth fix for December update - #23 by Den4t
In short, I ran:
pactl list cards
Then indeed, the sink for my Bluetooth headphones was set to “headset_head_unit” and not “a2dp_sink”. After running:
pactl set-card-profile bluez_card.E8_07_BF_FB_F4_14 a2dp_sink
It worked! No lag so far.
This is pretty hardcore stuff, it would be awesome if this could be documented somewhere. I’ve spent quite some time browsing through all topics, I have no idea what everything means exactly, I just want it to work 
Thanks again @fzinken