Yeah I opened it up in the end, not to replace the battery as I don't have a replacement anyway, but just to check that the dead one wasn't leaking (I bought a Mac IIfx out recently that I'd put into storage in working order, and in the years it'd been stored it turns out the battery had leaked and destroyed the motherboard. So I'm super-careful about dead batteries now.) Anyway the one in the T3100e looked fine, but I physically removed it just in case. It seemed safer.
But yeah I have all the utilities. I put Toshiba's own DOS 5.0 on in the end, so that should include everything. And I downloaded Net Tamer and got the thing online via dialup connection.
Now I have a new toy, a T3200SX, which should give me better expansion options. I wanted to get an 8-bit soundcard for the T3100e, but the prices of those these days are just silly. But with a T3200SX I can just drop in the16-bit Soundblaster that I already have kicking around. The T3200SX is in nice condition; screen works fine, no hinge problems and it even has 5MB of RAM. Only problem is that the hard drive has gone on it and I'm not really looking forward to stripping this down far enough to replace that (if I can even find a suitable replacement.) The most annoying thing at the moment is that because the CMOS battery has gone, it always starts up with the hard drive enabled. Which means that every time I boot it I have to wait for it to spend about five minutes (or so it seems) trying to read a dead hard drive before it gives up and I can boot from floppy. So I think it's more important to replace the battery on this than the T3100e so I can leave the hard drive permanently disabled. It's a great machine though.