Device does not have internet connectivity, there are going to be several more like this.
There isn't enough flash to upgrade RouterOS

Free Space = 6.8MiB
routeros-mipsbe-6.39.3.npk = 9.9MB

I've tried transferring the file to root (memory?), but when I reboot, the file is deleted and the OS does not upgrade.

Device does not have internet connectivity, there are going to be several more like this.
There isn't enough flash to upgrade RouterOS

Free Space = 6.8MiB
routeros-mipsbe-6.39.3.npk = 9.9MB

I've tried transferring the file to root (memory?), but when I reboot, the file is deleted and the OS does not upgrade.

You've chosen wrong architecture. Also, is there a good reason why you don't want to use current stable version (6.40.4) for your router?
I suggest you to download current version.

The .npk file should be uploaded into root folder. It will be placed into ram, so there is plenty of free space. After that router should be rebooted and upgrade procedure will start automatically.
Also the .npk files are protected from changes, so there is a chance your file could be changed\damaged during transfers from internet or usb drive or whatever.
You can check its hash sum to be sure it wasn't damaged:
MD5 routeros-mmips-6.40.4.npk: 61aad1990820481e3c209cf938b4783d
SHA256 routeros-mmips-6.40.4.npk: 8c6ea12fa963a521ee4748372e22f5a88e757704dba566e2ebe4139359cd6e9e

Are you not able to upgrade automatically?
System->Packages->Check for Updates->Download&Install

RouterOS files must be uploaded next to "/flash" folder, not inside it. Files are stored in RAM and are applied during reboot process if uploaded packages are for same architecture which router use, are newer than installed version and if there are proper packages uploaded (for example, you can not upgrade if system or RouterOS bundle package is not uploaded).

A little clarification as follow on what Strods wrote.

When you click files and you get the files windows take care that you don't have selected any files or folders (blue bar) and click the "Upload..." button and browse to the firmware file you want to use.

try this here:

viewtopic.php?f=13&p=624891#p624891