Stans, Mavic and Bonty have tyre size/rim width charts that might be worth a look at (google image search) I’ve had 2.75 on 50mm Hugo. Not sure if I’d want to go a lot narrower than that. Depends a lot on the tyre itself. some will show a lot more sidewall than others. There might be some 2.4s that would work.
a spare set of 29er rims would be your best bet, I reckon. More £ than spare tyres, but it would be much more satisfactory! you could even have them set up in the summer with some big road slicks for occasional road/bridleway bashing
Some further reading on theatre shape:
http://theatreprojects.com/files/pdf/Resources_IdeasInfo_typesandformsoftheatre.pdf