I ended up biting the bullet and followed the manual install described here: https://wiki.alpinelinux.org/wiki/Create_a_Bootable_Device#Manually_copying_Alpine_files
This was actually not so bad in the end. I had to look up where syslinux was installed on my ubuntu system in step 6, but found it quite fast and replaced the right part of the command.
Now, the Pi boots Alpine Linux from the SD card, which I now have to figure out. In order to do this I have to read a bit more about the process. The instructions for this are here: https://wiki.alpinelinux.org/wiki/Raspberry_Pi
Update: when trying to set up Alpine properly I broke the SD card. So I will have to do it all again. Be careful out there, kids.
Giving it another try. I set up the bootable drive again. What I didn't do last time was to create a partition on the SD card that I can use to store any data. I tried a bit to do this in Alpine, but ultimately chickened out and did it in Ubuntu using a GUI. I followed this guide: https://wiki.alpinelinux.org/wiki/Alpine_local_backup#Saving_and_loading_ISO_image_customizations
Following this guide, I then created an fstab entry to mount this partition on boot, and use it for cache etc. This worked! Alpine Linux runs now, and I can edit any changes with the lbu commit command :) Root password is: (redacted)