You seem to have made up your mind, but I will toss this out, anyway: it appears from your post that the alternatives you are considering have come from looking at vacancy announcements. If that is the case, I suggest that you do a bit more research by speaking with people who occupy such positions -- the much-ignored "informational interviews" that career counselors adore -- to find out more about what the real-life challenges and rewards actually are. If you're already doing this, that's great.
do I quietly start applying for these positions, or do I let my P.I. know I'm unhappy and that I plan to start looking for alternatives?
First, make sure you are sure that this is the direction in which you want to move. Next, instead of applying to anything for which you might be marginally qualified, try to find a few positions that seem interesting and a good fit for your skills. Look into the employer, the location, the work, whatever is relevant. When you have a firm plan of action and a few good leads, that is the time to approach your PI about acting as a reference for you. At no time should you say you are unhappy
. It will not help you if you give the appearance that you are reluctantly looking for just any old job because you don't like working for this boss. You should give the impression that, after thinking about your skills and goals, the job change will provide you with the venue needed to advance your career and make a real contribution to whatever.
Seriously, the only reason to tell your boss you are unhappy is if you expect them to say, "Oh, no, I'm so sorry! What can I do to make you happy here?" If that's what you want, that's an entirely different discussion.