In this scenario, it's less about the damage you can do to the company and more about the damage you avoid doing to yourself.
Integrity is something only you can define for yourself. If you're fine with it, do what you want and live with the consequences (or lack thereof).
To your example, I don't eat Chick-fil-A, and I don't shop at hobby lobby. There's something to be said for "there's no ethical consumption under capitalism" but those two companies in particular, I find repulsive, even though they remain incredibly popular. I know my boycott doesn't impact them, nor does it stop anyone else from supporting them, but I feel dirty when I shop there, so I do not.