A USB drive that ships as a 32 GB drive wouldn't be a fully 32 GB drive,it would be somewhat less-as you said in your case.
In that case,a 64 GB one may work(you may also try it formatting)
In any case,you can also skip copying system files to the drive-then it would require hardly a GB of space that boots up to advanced recovery options which allows to restore a system image.

Create a system image and save it to either a Hard drive/DVD.A system image is an exact copy of system files on a running computer.
System Image - Create in Windows 10 - Windows 10 Forums
When required,use the recovery drive to restore the system image.
let us know if there are any other questions.