I thought the same, but they are pretty unreactive compared to a benzylic proton, especially hexane.
I did this reaction a bunch of time with difference solvents trying to monobrominate tosyl chloride. I never had any side products like that hang around. The reason I switched around between CCl4, cyclohexane, etc was because I kept getting 10% of the dibrominated tosyl chloride, and separating them was very difficult (in the end, nigh impossible). I ran columns, recrystallized carefully from heptane, no luck.
One thing I just saw, you want to do it with 254 nm light, I used AIBN or benzoyl peroxide and heat.
Another solvent to throw out there for this, chlorobenzene.