Long time no talk. I recently came back to this issue because I was determined to get the Nvidia 5500FX card I have working, and I am happy to report that I finally have achieved stable 3v3 with an AT power supply using a custom 3v3 board based on a LT1584-3.3. I used two 220uF bulk poly caps, a 10uF tantalum, 22uF tantalum, a 0.1uF ceramic, and two ferrite beads for filtering, and was able to get down to 90~130mV of ripple on the scope. Couldn't get it to work on 12v rail, but was able to get it to work on 5v rail without issue.
The Nvidia 5500FX I am happy to announce likes this setup much better, but the heatsink I am using (a 50mm lengthed one) gets really hot. Definitely need to properly cool these linear voltage regulators if you go this pathway. I definitely recommend instead soldering the wire off the ATX line rather than going to the trouble that I have with an AT power supply, alas. It's done.
Thanks for all the help, I've done a lot of studying and learning electronics to get to this point and feel like I've come full circle now that I have a working 3v3 powered PCI port.