You need a specialist ----a property management specialist, that is. With four pets, you need to locate a sympathetic landlord. If your budget is flexible, then I recommend searching for a house versus an apartment. Near to Capitol Hill; try the Madison Valley, Madrona, or Central Area. You could also try Fremont, Ballard, and Wallingford which are west of the University District. All these areas offer more rental houses rather than apartments. Good luck!