In the end, I finally solved this problem by buying a hard drive drawer/caddy.
The drawer with the hard drive doesn't need to be removed to disable the drive - a lock on the front can be turned to the off position. At the moment, a quick reboot is all that's needed to enable/disable the extra drive. If I enabled AHCI in my BIOS (usually associated with SATA RAID, I think) I could even enable/disable the drive with the machine powered on, a lot like external hard drives. The reason I haven't enabled that - although Windows 7 has AHCI support built in, I dual boot and Windows XP needs drivers, and I haven't got around to installing those yet.
[EDIT I did try installing the drivers in XP, but the only way to enable AHCI in the BIOS is to enable RAID. When I did that, I discovered I'd have to reformat all my hard drives - my main system drive included - to use RAID. This hasn't yet been worth the hassle of re-installing everything again. I consider it a bit freaky that just in order to support powering a single drive on/off while the system is running, I have to not only reformat that drive for single-drive "RAID" but also the system drive which will always be on anyway. Still, it's probably a limitation in my BIOS rather than an inherent issue with AHCI - Asus/Asrock has wrongly assumed that the only possible reason to power on/off an internal drive while the system is running is when you're using RAID.]
[EDIT I now have a new motherboard (Asus ASUS M5A97 Evo R2.0), and AHCI works fine in both Windows 7 and Windows XP. There's a BIOS setting (per internal SATA port) to enable/disable hot plugging. With that enabled, I can "eject" a hard drive in Windows just like a flash drive. No reformatting needed. Works well for eSATA as well as for this drawer.]
On the noise issue, the drawer has a small and noisy fan. This is fairly easy to disable (a cable can be disconnected), but my drive gets slightly warmer than I'd like that way. So with the second hard drive powered on, my system is noisier than it was before. But that's quite rare, and with the drive disabled (even with the drawer in place) my system is quieter.
I bought a second drawer so I can swap in another drive, but I don't have another drive to swap in at the moment - mostly this is just a way to get a hard-drive on/off switch on the front of my case.
I have no idea how this particular caddy compares to others, but using a similar caddy may just be a workable solution for other people too.