Asking PI I barely interacted with for letter of rec? I'm a rising senior and will be applying to PhD programs this fall. I'm wrapping up a research project I was doing this summer at another school (basically an REU-like program) and the school is at the top of my field. I learned so much this summer and was able to accomplish a lot with my project. I was mentored by staff scientists and basically worked with them the entire summer. The PI of the lab is very hands off and hard to pin down. I met with him once in the beginning of the summer and have a meeting with him this week as I finish up. I was planning on asking him to write me a letter of rec then. The staff scientists said that what will probably end up happening is they write it and he'll sign it/co-sign it, but I need to ask him myself. I'm just a bit nervous/paranoid he'll say no so if anyone has any advice/words of encouragement, I'd appreciate it.

I understand that letters should come from people who know you well and I plan on my other two to be from professors that I am close with at my school. Given that I attend a school where there are limited research opportunities, I consider it to be integral that I get a letter out of this experience that speaks to my capabilities as a researcher. Based on the advice from the career office at this visiting school and other lab members, the letter should be coming from the PI/professor. The scientists I work with will contribute to a strong recommendation - they themselves acknowledge that it just wouldn't be as valuable coming from them. I guess my nervousness has to do more with the fact that here with big labs and big name PI's everyone says this situation/procedure is the norm and coming from a smaller school I'm not as used to it.
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.