I have seen the ideas of Alderson Disks around Sun-Like Stars. It would be utterly massive to say the least. Would an Alderson Disk be more 'possible' around a White Dwarf Star, how large would it be around an average White Dwarf Star? Assuming this Disk has an Earth like atmosphere, how would such a thing look and act?
First, you want a very old, very cold white dwarf. One such star (but you don't want it because of the neutron star nearby) has a surface temperature under 3000 K ... this is called white hot, but the temperature would put it firmly in the category of a red dwarf, if it weren't a white dwarf.
Such a dim star can have a "habitable zone" less than 1% the size of the Sun's (150 million km -> 1.5 million km or less), reducing the size of any "disk" with area by the same amount as the luminosity (over 10000-fold). Even more importantly, stars cooler than the white balance setting on your monitor shouldn't subject you to UV or worse radiation. Of course, that's still over a million km in radius, so I wouldn't book a trip just yet.
To come as close as I can think to plausibility, I'd suggest you postulate a natural ring of debris around the dwarf. You would like to condense some of that into a sturdy artificial moon, but your ring is somewhat closer to the star than the habitable zone. So you make it into an edge-on disk so the amount of light it absorbs is less, giving it a habitable temperature. With one of these you make a small gap in the rings over time, using up all the material in the gap. You can spin the built habitat to provide pseudo-gravity. Then you make another similar structure a little further in or out. As your collection grows, you loosely tether these objects together in a higher order frame with transport links, moving them each toward your optimal orbit. Eventually you have enough that you can call them collectively a thin or partial Alderson disk.