Maybe this person doesn't understand the proper care Bettas need. Sadly, petstores often give people bad advice. You might could print out a sheet listing all the care a Betta needs. Or, ask her straight out if you could have the fish. Maybe do both..?
I would be tempted to use the "your fish died" idea, however they may end up buying another fish. You wouldn't want anymore fish to be kept in those conditions. So, I think you do need to let them know about the kind of care the fish really needs.
Maybe just start out the conversation something like this, "I have been admiring your Betta fish and it got me interested in learning more. After doing some online research, I found out that..." And you can maybe have a printed fact sheet ready to hand them at this point. If they seem unconcerned or uninterested, maybe then you could ask if you could have the fish. If they say no, there isn't much more you can do. But at least do try to inform them of the care the fish needs.