There are a few large metros that have children's hospitals with less enmeshed academic affiliations (Dallas, Denver, and Atlanta for example - though each of those situations are unique in their own way), you might be able to get a set up like this. Of the 4 such individuals I've seen doing this, 2 have been PEM (makes sense given the shift work nature and types of complaints seen) and 2 have been ID.
Additionally, every city I've been in during training and as an attending has had independent GI practices not part of the academic machinery and so I think that would also be a potential avenue towards this idea. Not sure what it is, but GI seems to draw those with a greater independent streak I guess.
I'd think you could also do it in Peds Neuro, but most neurologists are already so busy that it may not work very well.
Lastly, I think there's a possibility in some smaller communities to make it happen. What many trainees don't necessarily recognize is that moderate volume NICU's with even just mid-level acuity can attract a constellation of pediatric subspecialists (GI, Neuro, Cardiologists, Endocrinology) simply to serve the needs of the NICU. It may only be one or two physicians per practice/specialty, but it can sustain some people especially if the cost of living is low. I've done locums in a couple different places where I've been surprised by the variety of specialists available to me, far away from the ivory towers. So in such a location you may be able to augment your subspecialty patient panel with some general pediatrics.
Now, all that said...If you are looking at entering a fellowship, you should recognize that probably 90%+ of the available jobs out there will not allow you to do anything remotely like what I've described. If you can't see yourself following the more typical path into academic medicine sans primary care, then you probably should stick with primary care.
The choice is somewhat similar to what I tell medical students considering Med/Peds...will your professional soul have a hole in it if you don't get to do both? Or is one just a "nice to have"? In this case, which one do you really need to fulfill yourself professionally?