Yes, getting the rear wheel off of a roadster really is a giant pain. It's not just you. Are you totally sure you can't patch the tube without removing the wheel? Try that first, if you haven't.
If not, then you'll probably need to take the chain off (at least, that's the case with my Raleigh DL-1). If there's no master link, you'll need a chain-breaker tool. If it's not the chain that's restricting it, but just the mudguard, then you've got two options: 1) take the tire off the rim (which you're intending to do anyway), if you think that'll give you enough clearance to slip the rim past the mudguard, or 2) detach the mudguard (which may be held on by the same bolt that holds the seat stay to the chain stay; don't be scared).
Sorry that there's no easy way to do it. The only consolation is that this is not a precision machine, so you can pretty much just attack it with a spanner -- you're unlikely to hurt it!