Asking PI I barely interacted with for letter of rec?
I've said it before, I'll say it again: writing a letter of recommendation for a student you know well is easy, it's a treat, and the letter you write can be GOOD. Writing a letter for a student you barely know is difficult, time consuming, and ends up sounding generic and forced. As someone who reads recommendation letters, it is very clear when the writer doesn't know the student.

I don't think them saying "no" is your problem here. See, even if they say yes, if you have this person who doesn't know you write your letter of recommendation you risk it sounding like a form letter which, in turn, won't distinguish you from the pile of other applicants. Unless the underling that they have write the letter really knows you and your potential, the letter is unlikely to be great. The box will be ticked but it won't be a meaningful tick.