Options is either shrink down / partition, either /home to make more space for /var. Is it possible to do it without full reinstall and/or losing existing data? I suppose I need to boot from live disk, but what then?
Shrinking down / would probably be the better choice, but that would mean having to move the start of the /var partition, which includes moving the data around. That’ll take a lot of time for 80GB, but it’ll work. You should be able to do this in both gparted and the kde partition manager, although personally I found gparted works better for this. Grab the manjaro gnome live image and boot into that, since the partitions need to be unmounted for resizing.
If you had used btrfs, you could’ve used one partition with several subvolumes instead, which would’ve been easier. Another option with ext4 would be to have used a LVM layout, but that can have an impact on performance. Both of these are too late now though.
GParted being used from a live ISO without any partition being mounted is usually pretty safe. Nevertheless, backup all data you don’t want to loose.
Shrinking /home looks to be the smarter approach as your / partition is already kind of small. In detail: Shrink /home and then move it to the right. Then you can expand /var. Compare UUID’s, they should not change.
If you use /etc/fstab to mount your partitions make sure to use the final UUID’s, then you should be fine.
I booted up Manjaro Gnome with gparted and without any problems or data loss shrinked down / and resized /var. Everything went smoothly and because of fast NVMe disk, the whole operation took 3-4 minutes.