I don't think it does at all -- I subscribe to your theory, that E-space was one of a number of alternate dimensions accessible during Five's time because other Time Lords were alive. Also, RTD has never made consistency with canon an important part of the new series -- in fact, I'm pretty sure he said during the "Doomsday" podcast that he wasn't interested in following the old continuity -- so that's another argument against any connection between E-space and Pete's World.

That said, DW canon is nothing if not wibbly-wobbly, so your husband's theory could prove to be right someday, depending on the whims of the production and writing team.