I recommend the Automated Fix because it includes the ATI rendering patch (tombati.exe) as well as improved cutscene options. The retail fix also uses tombati.exe which provides a windows executable using 3D acceleration (you can render at 1080p for example). The GOG and Steam versions both use DOSbox and software rendering.

Did you tried these?

http://www.glidos.net/tombati.html
https://github.com/ata4/glrage

Ok, for what it's worth my discoveries on trying this:

Glidos - Tried this on my XP rig with original CD, looks great especially with the texture add-ons. Unfinished Business was a little tricky getting to work as had to play around with some settings.

Automated Fix (with GOG TR1) - actually tried this on my Windows 10 rig at work, again looks really nice but the install would quit because the "audio codecs" couldn't be found so for the time being just went with the original setup w/o the audio upgrades. If anyone is familiar with this let me know how to fix it please, may be a matter of just downloading the codecs from somewhere.

I use a version once distributed by Cenega in Poland as part of a bigger bundle (TR1 through TRU, which was all there was at the time in one box).
It comes with DOSBox 0.73, which I update to 0.74 or SVN build for few more fps (or placebo effect).

Getting 3D acceleration is a bit tricky - on my WinXP laptop I use above-mentioned version with Voodoo Rush patch, appropriate build of DosBOX (Daum - yeah, I know...) and dgVoodoo 1.
On Win10 similar setup but with nGlide instead of dg.
UB works too.

I never tried to play ATI version.

It took a few years of waiting, but now there are two new options. Tomb1Main and OpenLara.
For Windows XP compatibility see my notes at VOGONS github.