Your best bet would be to join a group/hospital practice that is looking for someone to do a mix of general neuro and sleep. Such positions are not unusual, although I suppose you may have to consider relocating. Generally you'd start out doing X% general neuro and Y% sleep, and adjust your mix accordingly as things change over time.
Doing general neuro and "interpreting sleep studies to augment income" is a bit trickier, if by that you mean working for one practice and reading studies from other practices. It's unlikely that some other group is going to just start giving you their studies to read, unless they are mind-blowingly over-worked, in which case they should just be hiring you full time as a sleep doc. Also, reading studies on a "piecework" basis for a practice to whom you refer, is, to my understanding, a bit legally dicey as it may be seen as a "kickback" arrangement ("send us patients and we'll give you a cut of the reading fees . . . ").
Finding a 100% sleep job is challenging, although not impossible, and yes, you are right that much of it is done by Pulmonary. The neuro/pulm division of sleep seems to vary somewhat regionally. Also, in general, private practice sleep seems to be strictly divided into "all pulmonologist" or "all neurologist" practices, whereas academic centers seem more inclined to mix the two into multispecialty sleep centers (which I personally think is the better approach).