One solution is to create a sub-volume just for the data you DON'T want backed up and then move-and-symlink the directories
For example, assuming all the files & directories you want to exclude (except for /swapfile) are in your home directory, which is either a sub-volume itself or part of a sub-volume for all of /home:
cd /
sudo btrfs subvolume create caches
sudo mkdir /caches/rainer /caches/root
# swapfile, presumably owned by root
sudo swapoff /swapfile
sudo mv /swapfile /caches/root/
sudo ln -s /caches/root/swapfile /
sudo swapon /swapfile
# BTW, instead of symlinking, a better alternative for this
# file would be to edit /etc/fstab to use the new location
# of the swapfile.
sudo chown youruser:yourgroup /caches/rainer
cd ~/
# Before proceeding you should stop all processes
# that have open files in the directories to be moved,
# e.g. web browsers, file viewers (such as epub or pdf), and
# anything else that uses the cache director{y,ies} and is
# currently running.
for d in .cache/ cache/ [Tt]rash/; do
mv "$d" /caches/rainer/
ln -s "/caches/rainer/$d/" ./
done
# you can restart your stopped processes now
NOTE: you may run into a problem with [Tt]rash/, depending on the file browser you use and how it deals with the trash dir being a symlink rather than a directory. If that happens see if you can configure your file browser to use /caches/rainer/trash (but that might not be possible because most will use one hard-coded trash dir per filesystem).
In fact, it wouldn't hurt to do similar for all programs that use ~/.cache and ~/cache - move-and-symlink is a time-tested method that works and has worked for decades, but that's no reason to assume that software devs won't do stupid and lazy things when they get the chance.