Midnight Commander, or one of its more modern clones like Double Commander or muCommander, have full keyboard access. On Midnight Commander you add a file to your selection with the insert key, and you use the function keys to perform actions. The others will behave similar. Midnight Commander is a text application though. You can readily install it with "apt install mc". The others you need to download from their website.
Edit: an alternative I just found (but took we a while to recall
) is KDE's Dolphin. You can readily install it with "apt install dolphin" and it is a full featured graphical replacement for Caja. Once you have it installed and opened, go to Settings > Configure Dolphin, select the General tab on the left side (you may have to scroll down) and make sure "Show selection marker" is marked. Now every time you move your mouse cursor over a file or folder icon, in the top-left corner of the icon will be shown a clickable + or - icon. Click the + to add the file to your selection. Click the - to remove it. No keyboard