Installing on the drive
"could" lessen you chances of recovering files from that drive.
It is always better to recover files from a drive while booted from another drive. Sometimes this is not feasible.
If the software that you download can be run from a memory stick, I would go that route.
You will need to change your browsers preferences to reflect the new download location, the memory stick, before you start downloading.
Safari does not give you a choice as to where the download should go.