kidbayes

I don't think you can go wrong with the advice to "take more math", but I also don't think you can go wrong with the advice "make sure you don't do poorly in math classes."  Taking this class will, in all likelihood, not make or break your PhD admissions but getting a B or C will hurt.
Since you already have some analysis background, "doing analysis" as it relates to stats/ML is essentially using the ideas of convergence etc and applying them to probability/algorithms problems - you'll have to learn these things in the context of statistics anyways and you'd be better off just studying them in context.  For example, you'll take stats classes and learn things about convergence in probability -- you'd be better off learning about convergence in probability directly than taking a bunch of real analysis classes in "preparation" to do statistics stuff.  (of course, there are exceptions and if you want to do extremely theoretical research and go to UChicago or something, you probably want to take more math, but if that were the case, I think you probably wouldn't be asking this question)

As someone who applied this cycle, I would encourage you to take as much math (specifically analysis) as possible.
I think that, in general, applicants to Statistics PhD programs are coming in with more and more math background every year. Since the GRE/Math GRE are being phased out of admissions, the only way the committee can assess your math background is via your previous coursework. In fact, one of the professors on the admissions committee of a school to which I was admitted told me that every year, they are looking to admit students with stronger and stronger math backgrounds. 
And admissions aside, taking more analysis will ease the transition from undergrad to PhD.

I go to UChicago (for undergrad), and I've been admitted to top 5 biostats programs with B range grades in (accelerated) real analysis (all A's or A-'s but one in my remaining math/stat classes though).  I would just take it for a quality grade and wouldn't worry about it too much if I were you.  Many adcoms would assume that a P means that you were on the verge of getting a C range grade in the class.

I'd just take the letter grade. Some professors I've spoken to say that adcoms may assume the worst (B- or C) when you p/f a class, and real analysis is arguably the most important class on your transcript. 
As to how much it will hurt? Probably contingent on your performance in other math classes and the rest of your profile. For reference, I got a B+ in RA with mediocre grades in some of my other math classes and was still fortunate enough to get into some T30 programs.