Well, switching hosting company was pretty straightforward, though I'm not sure how quickly the "real" domain name will be active. For the time being, the site's reachable at http://www.rsaunder.myzen.co.uk.
My new hosting company uses cpanel to administer the account, which appears to be an improvement over the previous setup. The pages seem a bit quicker to open, though it remains to be seen whether I get 500 internal server errors.
It was a bit of a fiddle transferring the Firestats hit logging system. The main part of the site was easily copied over using Joomlapack, to whom I'm grateful!