Cards with external power connectors are certainly more stable on old motherboards, but they still draw some power from AGP. So it might not help you with very low current limit. You can also try to find 5900XT.

There is official Asus rework for early versions of P2L97 which included soldering wire directly to a capacitor and unsoldering voltage regulator leg.

It would depend how the particular card is designed. The auxiliary connector won't necessarily be powering the same components as the AGP slot does, so it's possible some cards might still need to draw a lot of current through the AGP.
In actual practice though, I don't know which models would be better or worse in this way.

The 6600GT is a bridged card - the GPU is natively PCI Express and it has a separate bridge chip to convert it to work on an AGP interface. I don't know if bridged cards will work on a 440LX - they don't work on everything and the 440LX would have been seen as irrelevant by the time those cards were made.
I don't know much about the 6200, but I wonder if it's bridged also?

The Geforce2 MX and many of the 4MX cards are light on power. I'd be surprised if those wouldn't work - a lot of 2MX cards (especially the non-retail ones found in prebuilts) don't even bother with a heatsink.

In case you want to modify the board, here is the rework procedure Asus put out back then:
https://web.archive.org/web/20010401023659/ht … tnt-rework.html