I think the Ba'al Shem Tov had to have existed in some way (even if it's not the way we've been taught), as I doubt an entire branch of Orthodox Judaism would base itself off the teachings of a ficticious figure. Let me ask you this - if the Besht didn't exist, who made him up?
From a historical standpoint, saying he didn't exist is futile. We have books which belonged to him (such as his siddur), the shul where he taught (which was sadly destroyed by the Nazis, y.s. and has been replaced with a replica), and his grave. Also he is recorded in the 1758 Polish census.
I think with the above evidence, it's pretty clear he existed. I think the question is not "did he actually exist" but rather "did he exist the same way he's presented in all the stories". And now here's where the following quote comes in handy:
"If you don't believe any of the stories of the Besht, you're a heretic, and if you believe all of the them, you're a fool." - Rabbi Shlomo Rabinowicz, the first Radomsker Rebbe
The variation between each of the stories is where the problem arises. In some stories his entire backstory changes. I think we ought to look for which stories come from authentic sources, and throw out the rest. I think a lot has been embellished throughout the years.