Realistically, I dont think that chair lends itself to much in the way of modification or outdoor use at all. I presume you mean for loose surfaces. You will get better grip if you move the C of G BACK.
It uses 12.5" x 2.25 tyres too which are kind of mobility sized only. And no matter what you replace them with there isnt much to be gained as they are too skinny/small. Any of these
http://www.ebay.co.uk/sch/i.html?_nkw=12.5+x+2.25+tyre should fit. As long as your rims can take air filled tyres.
Otherwise it means fitting different wheel rims for wider/bigger tyres.