What corporations can I run missions for and avoid losing standings with the major factions?

Solution 1:

Eh I wouldn't say that Sisters are neutral, or at least they weren't a few months when I played last. They give you gallante with minor bonus to minmatar.

Can't say that I've ever ran into factions that were neutral. If your'e trying to keep your totals with factions from being lopsided, only thing I could suggest would be grab a minor corp (aka no navies) with decent lvl4's in the amarr/caldari, and the gallante/minmatar camps and mission back and forth.

If you decide to do so, be carefull to accept only the missions where targets are pirates, or non-faction ships, else it would be a huge blow for your standings.