Probably the best way is to go for specialist forum hosting.
There are a variety of hosting companies who support IPB (which I'd suggest we stick with). The software itself is around $160 with yearly maintenance, but we can avoid that by going for a hosting plan that includes it.
Invision themselves provide hosting, and there are several other companies who specialise in forum hosting (which is going to be the way to go as a specialist company is more likely to have their IPB installation configured so as not to cause CPU overhead problems). Costs start fairly low from just a few bucks per month. We just need to make sure that we sign up with a host who will allow us to upgrade as the site grows. As and when we're ready to go, I'll do a little research and come back with a definite proposal. All we'll need then is a domain name (or to transfer the original, if Blade is agreeable) and we're in business.
I can organise all of the above without any problems, and am happy to do so as it's very straightforward. However, if Blade does allow us to move the old database over, I'm going to be struggling to find the time to do the transition as work and other commitments are taking an awful lot of my time at the moment. It's not just going to be as simple as copying the old db over - it's bound to need cleaning up and tweaking for full compatibility with the latest version of IPB.
I'd suggest that the first thing we do is try to get a definite response one way or the other from Blade. At that point we can decide how to approach this.