I have stumbled across the reason my BaseCamp session seems to hang with just the colour wheel rotating for a few minutes at a time. It's down to the fact that Time Machine is busy updating it's backups. When it does so, it mounts the Time Machine as a volume under Finder. BaseCamp picks up that a new volume has appeared and tries to do something clever like look for maps. The result is BaseCamp hangs until Time Machine is finished and dismounts the volume. This is behaviour is new to v4.1.1.
The work around it to stop Time Machine manually (if displayed, click the Time Machine icon in the menu bar at the top of your screen and select "Stop Backing Up"). If you have the default time machine settings, you'll find yourself doing this once every hour while you are running BaseCamp!
I reported it to Garmin support and they just said:
"Unfortunately the configuration of the hardware of the Time Machine may
conflict with our software as it is a removable/virtual drive.
Unfortunately we have no control over the Time Machine configuration the
only solution we could offer is one you have already tried which would
be to temporarily disable this service while using our software."
Hope this helps someone, and Garmin sort something out to either recognise (and ignore) time machine volumes, or allow users to specify which volumes to ignore.