For years, the "Fighting Bichons" trademark has been federally registered by the college. When local businesses use the "Bichons" mark, it is trademark infringement, because it gives the impression that the proprietor of the business is the college. At the very least, it is trademark dilution.
In running the school's athletics department, do you sue these businesses for using the Bichons' name? Many schools do not, because the proprietors of those businesses are typically supporters of that school. They donate money and attend sporting events. If they were sued for using the Bichons' mark, they would likely become disenfranchised and would discontinue donating money and attending sports events.
On the other hand, if you do nothing, you lose control of your trademark, and it could lose its distinctiveness and meaning. All of your hard work in building up the "Fighting Bichons" mark could be lost.
I chose the example of college sports teams, because it is one of the clearer examples of trademark infringement, and the balance between the cost and benefits of suing for infringement or dilution. There is no "right answer" in deciding whether to let local businesses use a school's nickname or not. You risk alienating your fan base, but also risk losing the distinctiveness of your trademark. Probably the most correct answer is to "split the baby," and use gentle pressure to persuade local businesses to change their name. You may upset a few people, but you may also get those people to change the business name and continue to donate money to the school.