Does anyone know if there's a way to disable, or at least shorten the duration of, the little tooltip that shows in the bottom bar on the Linux version of GIMP?

That isn't a "tooltip" - it's an "info" message that is produced automatically whenever you do Edit>Copy (or a script or plug-in invokes the equivalent procedure).
AFAIK you cannot disable the message, but you can re-direct by activating the Error Console dockable dialog.
ie.
<Image> Windows > Dockable Dialogs > Error Console
This will create a new tab in the upper-right window in Gimp, after the "Layers", "Channels", "Paths", etc tabs, and all info/debug/error messages will then be shown on that tab rather than on the bottom line of the main Gimp window.

Thanks! That fixed it.

It was getting annoying, since I had to wait for it to disappear before I could read the status on how many pixels I had moved a selection.

Great!
Just bear in mind that, in future, all error messages will be shown on the Error Console rather than the bottom line of the main Gimp window - usually if an error message is produced by a script or plugin, Gimp will automatically show the Error Console tab (rather than Layers, etc) - but sometimes it doesn't. So if a script or plugin appears to "do nothing", check the Error Console tab.