I tried replacing the cat.svg with my own by adding $wgTimelessBackdropImage = '$wgResourceBasePath/path/to/my/logo.svg';
to my LocalSettings.php; but then nothing shows up. I've replaced the cat.svg in the skins/Timeless/resources/images
folder for now, but there has to be a better way.
Topic on Skin talk:Timeless
PHP only evaluates variable names in double-quoted strings. Thus, you need to write $wgTimelessBackdropImage = "$wgResourceBasePath/path/to/my/logo.svg";
instead.
It worked, thank you!