A GF4 Ti4200 is compatible, affordable, readily avaliable, has decent DVI and VGA output and will still be bottlenecked by any Coppermine PIII.
I have tried the card with other Apollo Pro 133A boards and never had a problem. Still your milage may vary. Everyone has different experiences with VIA chipsets. Some say they're all trash, others never had a problem.

The question is why the board seemed picky. It's a solid enough universal AGP design, so assuming you installed Via 4-in-1/Hyperion drivers correctly, the problem is something else. The obvious difference with that 9600XT vs anything current when this board was designed is its power draw, both in terms of sheer current and the lines it does it on. It could be your board not being able to deliver it, or the power supply you are using (any reason you don't list that in specs?) The fact that the 9200SE does work indicates that it's not an AGP 8x incompatibility as such. If you are using an old, period-correct PSU (or a new crap one with weak 5V line), it's worth testing with a different PSU. If that doesn't change anything, it is the board that's limiting you.

That aside, I'm not really convinced this system is going to do what you want - the video card aside, this is a 1999 build, regardless of GPU I don't see it playing 2002 games adequately, that P3-550 would be too much of a bottleneck. A Radeon 9600XT really wants a >>1GHz CPU. Did you really intend to run a 9600XT on a 1999 system as a kid or would it have gone into a 2003-era system?

For a 1999 build I'd go for a 1999 card, and the best in terms of compatibility would be an early GeForce (as nVidia's DOS VESA support is a bit better than ATi's) or if you can find/afford one a Voodoo3-3000. If you just want a fast card so that it's not the bottleneck, the trick seems to be to get as much performance with as little power draw as possible. Maybe you could look at things like a GeForce 6200 AGP.... do note though that somewhere past the FX-series nVidia's excellent DOS VESA support was dropped, so no DOS there.

The power supply I'm using is a 600W model from Ultra that says it's rated for 30A on the 5v rail and I had the same issues with a different power supply as well. The 9200SE is actually a PCI card, but the Rage Pro that's in it is AGP and works fine. The CPU is definitely going to be upgraded to a 933 or 1GHz chip at some point in the future and I'm just trying to use a 9600XT because I got that card for free, though I likely would have bought a 9-series card of some sort as a kid because that was what was available in retail stores at the time.

I did not try the VIA 4-in-1 drivers and I'm using the ones that came with 98SE. The system will not even post with the 9600XT in, though. Am I missing something about the drivers? Do they alter some values in hardware?

Would an FX 5500/5200 be a reasonable card? I might be able to get one locally.

A GeForce 3 Ti200 can be had pretty cheap in OEM form with passive cooling and is a good match for a high-end PIII. Search for the dell part number: 04n857. It’s a late 2001 card and will be a bit faster than the 9200 SE with its 64-bit memory interface. A Ti 4200 is faster than the Ti 200 and can also be found cheap using dell part numbers (03w694).

I’d skip the 5200 because so many of those cards are 64-bit memory interface models. The 5500 should be better, but personally I’d still go with a Ti 4200. DX9 support is pretty useless with the low performance of the NV34 chip in the 5200 and 5500.

I agree that a Ti 4200 or GF 6200 AGP would be probably the best for your board and cpu. The updated VIA GART's for Win9x is better than the one included in Win98, never had an issue with most versions though v
Also there is a reg fix to ensure the AGP is running in 4x mode under Win98 (which defaults to 2x due to older GART) you can still get from Asus..

Early Via AGP compatibility was definitely flaky. My Via 133a and 133t socket370 boards are guaranteed to crash with some early AGP cards like Trident 9850 or Number 9 Revolution 3D. The same cards work ok on Intel chipsets.

For the OP, definitely try installing the Via 4 in 1 driver. And maybe stick with Universal AGP cards, those with two notches in the AGP connector. That means Ati 9800 or older, or Nvidia 5000 series or older.

Obviously it might help to pick a card that doesn't consume a lot of power, like a Gf3 ti200, or Gf4mx, or Radeon 9000/9200.

As others have said, going with cards that have low power requirements is a good start. A Voodoo3 2000, a GeForce2 MX400 or a GeForce4 MX440 should all be safe bets.

If you limit yourself to games made in the year 2000 and earlier, and don't mind gaming in lower resolutions, all of the aforementioned cards will serve you well. Your CPU will likely be a bottleneck for anything newer than that, so a more powerful card might not do you much good.

Geforce FX5900U. Has early dx compatibility that radeons ansd latter Geforces lack, and uses external molex power so agp power draw not an issue. Excellent image quality and support for fos games, FSAA etc, just drops the ball with dx9.

Is the VIA chipset on this board a CD or CE revision? The later CE is a lot better when it comes to AGP compatibility.

I think replacing VIA with i815 would be more appropriate, it saves you from troubles searching for videocard. Coppermine boards still cheap and plenty.

As a few others have pointed out, a 9600XT is a bit much for a PIII-550 and would be much better in a P4 or fast PIII coppermine system. I currently have the 9600XT installed in a PIII-1000 and it can play Halo: CE pretty well in that system, but that system is a Compaq DeskPro and it has a few BIOS imposed limitations that I don't like. I do have a couple of decent P4 boards around, so I might consider a build using that card in one of those.

I think I may just keep this system how it is at this point and find a good P4 machine to dual boot 98 and XP on. Most of the value in keeping this system for me is that it has ISA and a little bit of sentimental value as it was one of the first computers I used.

Thank you all for the advice, I will still keep my eye out for some of the cards mentioned here in case they go up for sale locally.