I remember having the exact same problem with my 220. Though my problem was running the game in Windows. Once I ran it in DOS, it worked fine. Also make sure you're running the correct version of the Mystique patch. On the retail CD, there's two folders, Myst4MB, and Myst2MB, the one you want is the 2MB as that has 220 support (it does support 4MB cards, the folder names are misleading). The readme says it's version 1.03 (should also say this when the game closes).

My Tomb Raider CD contains only one folder with Mystique patch, under D:\PATCHES\MATROX. I think it's not a retail CD, it's from a CD-Action game magazine.
Other stuff in the PATCHES folder:
3DFX folder
POWERVR folder
RENDIT~5 folder (visible as Rendition under Windows)
S3 folder
VOODOO folder
TOMB.ZIP
TR_PVR.ZIP

I attach all the TXT files for additional information. I would copy paste them into the post, but I don't know how to make them folded and I don't want to make mess in the post.

The MATROX folder contains following files:
TOMB.EXE
TR_MYS~4.ZIP (visible as tr_mystique.zip under Windows)

The TR_MYS~4.ZIP archive contains following files:
TOMB_MYS.EXE

The TOMB_MYS.EXE is a self-extracting archive and contains following files:
TOMB.EXE

The TOMB.EXE from the MATROX folder is 1222303 bytes in size.
The TOMB.EXE from the TOMB_MYS archive is 1026747 bytes in size.
I have tried using both of these executables and they both produce the same results, described in my previous post.

Edit: One more maybe important thing, text which is displayed when I exit the game:
The TOMB.EXE from the MATROX folder displays:

1Tomb Raider (c) Core Design. Sep 04 1997 03:47:06
2Matrox Version 1.03

The TOMB.EXE from the TOMB_MYS archive displays:

1Tomb Raider (c) Core Design. Feb 20 1997 18:13:26
2Matrox Version 1.01

Yeah, the one in your MATROX folder is the correct one. That's the one I use. It should also be the most updated one. Odd that it still doesn't want to work. I just tested mine again under MS-DOS 7 (from 98SE) and it was fine. There should also be a 3D scene before the EIDOS white screen that uses the same mode as the game does, when not under a cutscene, saying Matrox in 3D letters. I wonder if it has something to do with FreeDOS, as I see you're using that. I can't be sure though. I've also never had to use UniVBE for this, it doesn't need it. Do you have a different OS you could try it on by chance? And maybe check to see if the Video BIOS cacheable setting in your BIOS is disabled, it can cause problems for some things.

And yeah, on a side note, software rendering of the game is very slow unless you have a beefy CPU. Even my K6-III+ at 400 MHz struggles a bit in 640x480. My Pentium III at 866 MHz though can run it at a solid 30 fps.

The Matrox 3D scene does not appear, but there is black screen for longer than normal before the EIDOS logo appears.
The Video BIOS Cacheable option is disabled.
I'll try temporarily plugging in another hard drive and installing MS-DOS 6.22. If it won't work on MS-DOS 6.22, I'll try Windows 98. I still prefer FreeDOS, because it's still maintained, so I will disconnect FreeDOS drive for the time of testing. When I'm done, I'll edit in the results.

PS: I just got a Voodoo 2 (maybe this will work), but for a suspiciously low price, so it may not work, and I don't have the VGA jumper cable yet.

Edit: I decided not to test with MS-DOS 6.22 and go straight to Windows 98SE. Matrox logo appears and the game works fine, so it's probably something with FreeDOS that is causing the issue. But Windows also installed Matrox Mystique drivers. Maybe those drivers contain some important file that is necessary to run 3D games in DOS mode, like glide2x.ovl in case of 3Dfx cards.
I did the test with MS-DOS in command prompt mode, because the CD drive was not detected when I used the "Reboot to MS-DOS" option.

Edit: I installed the Voodoo 2 card, set the variables, and installed the proper Tomb Raider patch. Both the card and the game works perfectly in FreeDOS. I don't have the VGA jumper cable, so I have to switch the cable between cards manually (ATI MACH64 as the 2D card), but I think I'm going to buy the jumper cable soon.
My problem is solved, but the Matrox-in-FreeDOS issue is still unresolved. If anyone has a solution to that, I would appreciate it.

Edit: Update on the Matrox issue - I downloaded Matrox Simple Interface and tested the card using it. At first it didn't work and displayed glitchy stuff on the screen.
But then I booted into FreeDOS safe mode (bypassing some drivers) and IT WORKED! Now I just need to check which one of the drivers causes the problem, by disabling them one by one. Maybe it's all of them, and the problem is caused by them taking up space in conventional RAM, or cache, or something.

Edit: I finally solved it. The solution is: Don't use JEMM386 (FreeDOS 1.2's version of EMM386), or JEMMEX. HIMEMX can still be used, but it's not necessary for it to work. Case closed.

Hmm, that's very interesting. I wonder why those are interfering. To note, I tested with EMM386 on MS-DOS 7.1 and it was fine. Still don't know why it doesn't work in Win98 for me though, but it's not really a problem. Glad you figured it out, I would never have guessed the memory managers were at fault.