I'm looking at a higher model HTPC graphics card that can do some gaming (Oblivion and whatever else would run.), doing HTPC functions and operate with the stock PSU. So which of these 3 cards would be capable? I've heard of 5770's running on a 350W system, but according to what I've read, a 5670 or 5570 would be the much safer choice.
The PSU isn't crappy and certainly doesn't need to be replaced for an HD5670. Dell rates their PSUs for continuous usage rather than peak which makes them underrated when compared to normal PSUs. With 300w on the +12v it should be able to handle even an HD5770 just fine.
@OP What is the native resolution of your monitor?
For those resolutions I would suggest the HD5770. It really should be OK on your PSU. The card maxes out at 85w under normal usage so 300w should leave you plenty for the rest of the system with head room to spare.