Unfortunately, no. It always defaults to My Documents.
It's the simple things like this feature that could really help improve a product from a user's point of view. :-)
The location is established by the Environments setting in the OS. This can be altered, at the OS-level, but will be global for all programs working in that OS.
One could write an Environments variable CMD-line file, to be invoked and chosen at boot-up. The structure of this Environments variable will likely differ OS version to version.
Older OS's were not so tough, with things like this. With newer OS's ?
Before I did such, I would do a great deal of research on my particular OS, and look at the required structure and syntax, plus also set this up, so that I could make a choice at boot, rather like the option, when booted with F8 held down, to go into the OS as Administrator, a User, Safe Mode, or Safe Mode with Networking.
Good luck, but please be very careful and also make sure to have a safe backup of the Registry, that one could invoke, should things go very wrong.
You can save your share settings as a Quick Share. Then the next time you want to share your video with those same settings, you can use the Quick Share feature, which retains the "Save In" path. Quick Share doesn't stop to let you change a setting, so if you do need to change a setting you will have to go the regular route.
You can save your settings as a Quick Share just after the export has completed and before you click "Done" if I recall correctly.