Update: it didn't go smoothly
The Radeon 9000 card arrived this week so went to my dad's today to fit it. Since installing Mint on his PC, the screen has only gone corrupted a handful of times, only for a couple of seconds at a time, usually not long after boot-up. Everything else (other than mintupdate) has been fine. Nevertheless I opened up the case, fitted the card, changed the BIOS setting and booted up.
However it wouldn't get past the POST. It's an emachines PC -- the "e" logo showed, as always on power-on, but after it disappeared, nothing else happened: after a few seconds the monitor showed "No Signal" and then nothing else happened. I powered off by holding the power button down and tried again, and this time got the GRUB menu. I chose advanced options > recovery mode, and this time it started up, right the way up to a working desktop, albeit with a slow and flickery mouse pointer. Subsequent shutdowns and restarts were the same - booting would result in nothing at all, hard-powering off and trying again would let me start in recovery mode. I figured the open source driver wasn't handling it so tried the Additional Drivers program; this reported there were no proprietary drivers.
So I shut down and took the new card out and re-enabled the integrated graphics, and it worked as it used to. Strangely though I noticed some of the fading screen effects - like the screen darkening behind the shutdown confirmation box - were now working where they didn't before. So I'll monitor how it's running back on the Intel graphics. If I press on and try to get the ATI card working I'll start a new thread here.