In the end, I just went to a local computer shop with a decent warranty and picked up a card. It worked.
Then I got home and read this response by Bob M. to my Usenet query, which I’m now passing on for the sake of completeness:
DVI (and similar digital interfaces, or even analog types, for that matter) really doesn't have a "max. resolution" (pixel format) per se. The limiting factor is pretty much always the pixel rate, and as long as you can put together a valid timing for a given format which does not exceed the rate limit of the interface in question, the interface will be happen. For instance, a format of, say, 200 pixels horizontally by 10,000 vertically at 60 Hz refresh may be a silly thing to do, but since the pixel rate would probably come in well under 150 MHz it wouldn't pose much of a problem for most interfaces.
With respect to DVI specifically, the maximum pixel rate permissible (in the single- pixel- per- clock mode) is 165 MHz; this is sufficient to cover the VESA standard timing for 1600 x 1200 at 60 Hz (162.000 MHz pixel clock), even at CRT-like blanking times. Reducing the blanking will increase the active pixel count supportable quite a bit higher, of course. However, whether or not a given implementation can actually support the specified max. rate is not guaranteed; cable and connector quality issues may limit the max. rate that can be reliably used to something less than the spec. maximum.
Thanks, Bob! This makes sense to me at last!