Don't ask your PI for an LOR. I repeat, DO NOT ASK. It will be a bad letter, which is worse than no letter at all.
Whether or not the absence of this letter will be a red flag is questionable. I think it depends upon other aspects of your application. Is there anyone else who can write a research-oriented LOR for you? I ask because I've worked in several labs over the years, and just had major personality conflicts with one PI in particular. He did not write an LOR, but since I had other people I could ask, the adcoms were able to see that I was good at research and able to work as part of a team. I had plenty of interviews, and the absence of that letter in particular was only mentioned at one of them -- and I gave an honest answer: "My advisor and I had a strained relationship, and I didn't feel comfortable asking him for a letter." I figured that if they wanted more detail than that, they could ask (they didn't).
On the other hand, if this is your only research experience, then the lack of a letter will stand out more. However, it's really a moot point. You will be applying to schools without an LOR from this PI, and it may or may not hurt you. You can't change the situation, so try not to dwell and move on. You certainly won't be the first person to have something less-than-perfect about their application! In interviews, don't bring up the topic, but if you're asked about it, answer professionally and briefly. If they have specific questions about the nature of your relationship, they will ask you.
Personally, I don't think this will be a deal-breaker for you. It might be a bigger issue at a heavily research-oriented school, but otherwise, try not to stress. What's done is done.