It will be tough over Easter Week especially if you want < £300 including taxes and fees. Have you tried filtering for everything in your price range that meets your space criteria and then vetting them individually? I'd try vrbo.com and booking.com as well as AirBnB. If your husband still has friends/family in SF, it may be well worth his asking around for recs - the rules for non-commercial short term rentals in SF are very restrictive on how many days per year they can be offered, so you sometimes find people who'd give priority to someone they "know".
You have the right idea looking on the west side of the peninsula, inside the city but outside of the main "grid". There's often a variation of names used for different areas/neighbourhoods, so this is a bit of a "brain dump" including the neighbourhood name variations I can think of, just in case you might have missed some. Working north to south:
Presidio, Sea Cliff, Baker Beach, Lincoln Park - south of the Golden Gate Bridge, north of Richmond. These areas are mostly urban parkland, but you may find a few listings around the area.
Richmond you've tried - make sure you've included Sutro Heights and Vista del Mar (these may be labelled as Outer Richmond or by the more specific name), Little Russia.
Golden Gate - these are essentially either Richmond or Sunset - although "Golden Gate Heights" is mid-Sunset.
Sunset - this would be my first rec; it runs along below GGP as Richmond does above. If you see any listings for "Zoo", they're worth a look - it's on the outer edge of Sunset but good transit links. (I think there are planned outages on some or all of the L Taraval MUNI/Light Rail line, so check to make sure you have light rail near you that's actually running if it's important to you - buses are safe, clean, and reliable but they do run the risk of getting stuck in in SF's pretty constant, pretty congested traffic.)
Moving south and east: Parkside, West Portal, Cole Valley, Portrero Hill - these are closer in, so likely to be more expensive, but also likely to have 3+ bedroom rentals.
Lakeshore, Stonestown, Park Merced, Ingleside Heights - a bit further south but workable.
What I WOULD be very wary of is "deals" and "hidden gems" closer to the city centre (some cheaper areas are Civic Center, Mission, Van Ness, Tenderloin, Rincon, Little Saigon, and possibly parts of SOMA/China Basin). While the property may be fine and well-located/connected, the neighbourhood may be more gritty than you'd want under the circumstances.