Since my original post, I have spoken with support at Bluehost. They believe that I've done everything correctly but there is an issue with the Wordpress MU Domain Mapping plug-in (or at least my implementation of it), which they don't support. They believe that the basic issue is that I need to have purelylawyergifts.com mapped to purelylawyergifts.purelybrands.com (so that when someone types purelylaywergifts.com in their browser address bar they end up at purelylawyergifts.purelybrands.com), but they believe the plug-in is set to do the opposite.
I can't seem to add a screenshot of my current settings for the plug-in, but if I could perhaps this would help others to diagnose my problem.