So, my solution has been to make a "dummy" file, which has the root name I would like, but with no index number. Then I put this "dummy" in the same folder as my scans, like so:

- Screen Shot 2015-07-22 at 2.25.40 PM.png (245.06 KiB) Viewed 2313 times
So when I go to batch scan, I can click on the dummy file, and only that will fill in as the root name:

- Screen Shot 2015-07-22 at 2.35.49 PM.png (246.13 KiB) Viewed 2313 times
NOW I can hit 'Scan' and the files will be named properly.
So just to be clear, here are the steps I have to take to do a batch scan now:
1) Change the "dummy file" to the proper root
2) Click 'Batch Scan'
3) Open the folder that I told Silverfast to save the files into
4) Click on the "dummy file"
5) Click 'Scan'
And here is what it used to be, not too long ago:
1) Click 'Batch Scan'
2) Click on most recent scan
3) Click 'Scan'
I'm sorry to keep bringing up how things "used to be", but I do it because it makes it clear that it worked at some point! I don't know how software development works but is not simple just to look back at the old code and repeat the old process?
Thanks for your understanding.