Fionn, it's probably a bit too short notice, but what helps a lot when flying handicapped people, is a vacuum mattress.
We use these a lot for handicapped kids who's ability to sit straight up is marginal. They're basically mattresses filled with rubber granulate. You put the mattress in first, put the pax on it, and shake it about a bit so that everything sets well. With the help of a few people you fold the various sides and ends either against the body of the pax, for support, or away so it doesn't interfere with the aircraft controls. (Most often though, these pax are flown in the back.)
You then connect a (hand-operated or electric) pump to the vent and suck the air out of the mattress. Similar to a bag of coffee, this squeezes all the rubber granulate bits together and ensures the mattress cannot change shape anymore. This will give the pax excellent support during the flight.
The ones we have (our charity has three by now) are from RedVac.
RedVac Catalogue - Kohlbrat & Bunz - PDF Catalogue | Technical Documentation
The first one we got was the straight 200x70 (or so) mattress. We found that this one is too big for sitting upright. We placed a special order for a 150x70 or so mattress, with indentations for the shoulder belts. That's the one we use most often in a C172 or similar.
Another tip is something you've probably already figured out. Handicapped people are most easily flown in a high-wing aircraft such as a C172 or C182.
And yet another: Make sure you have a few luggage tags (with a bit of rubber band or so attached) available. When a pax is rolled to an aircraft in his own wheelchair, tag the wheelchair with the callsign of the aircraft. When the aircraft returns, you know which wheelchair to grab and roll to the aircraft.