Firefox’s free VPN will offer 50 gigabytes of monthly data, which is pretty generous for a browser-based VPN. A Mozilla account is required to make use of it, which isn’t a hardship (they’re free), but is a point of friction some may wish to know upfront.
You could try connecting an external keyboard/mouse to your main computer.
As for the backup one - how much space do you have left? You’d need between 4 and 6 GB to download a Linux ISO and around 2 MB for Rufus with which you’d build a bootable “live USB”.
If you don’t have even that much, grab WinDirStat to check what exactly is taking up so much space - maybe you can remove some of it.