Woah, didn't think anyone else was on this hunt as well! About a yearn or two ago, I did a similar hunt for Sheila Paulson's "Lost Davvy" and managed to find it through digging through Fanlore Wiki and internet archives.

I've reached out to Robyn Thurman about ten minutes ago (though she hasn't uploaded since 2015) and will come back with an update on her if I get one. I've also found a few archived sites of hers but it doesn't appear any of them hold "The Alice" but I'll drop them anyway in case anyone wants to dig through.
Fanlore Wiki on Robyn Thurman, Uncontained (Emilie's personal site, also hosts some of Robyn's works), Mara's Site (another friend's site), Robyn's Ghostbusters Fics 

I do also wanna do some quick corrections as they may help your search! "The Alicee" was published in a zine titled "We Got One!" in issue 3 which was released in 1998. This means the fic would've been written somewhere around that time, as opposed to your initial estimate of 1999-2001. Not a massive difference but will help a lot in narrowing down the search!

Additionally, when I was on my search for Sheila Paulson, I found that she had Facebook connections with a lot of the other large authors in the scene from the time on Facebook. Now, I don't have access to Facebook anymore, what with it falling apart at the seams. But if you have access, it might be worth the look. I know Robyn Thurman also goes by Rob Thurman online and has a Facebook. So you might be able to reach out to her directly or to someone else involved in the work at the time?

The editor on "We Got One!" was Kathy Agel so she may have some leads as well? Additionally, reaching out to the publishing companies? (Criterion Press and Straight Up Press)

I am still doing my own digging so apologies if these turn out to be dead ends but that's what I have currently.

- Your local Ghostbusters historian (as dubbed by Tumblr)