So, I thought for a long time about making some homebrewed spec tree to reflect spiderman's abilities, but then I decided, "meh, forget that noise" and just decided to work with what's already available, and I found the perfect combination!!
Martial Artist + Politico.
MA for obvious reasons, it gives you lots of ways to defend yourself based on coordination, and also improve your offense with coordination as well, which was always the "gymnastics" skill in my head. Plus, it gives you the Precision Strike talent line, which lets you inflict Crits of your choice, allowing you to take down enemies quickly, and with minimal effort.
Politico, is for the Scathing Tirade line, especially Supreme, allowing you to taunt your opponents as a maneuver, which still allows you to beat them up as well. And it's pretty thematically fitting too. Spidey very often will just bounce around, avoiding his opponents attacks, making them more ticked off with each failed attempt. This is the strain loss from the ST checks, slowly working them into a mindless frenzy. And then Spidey will either sucker punch them into unconsciousness, or with enemies like the Rhino, let them knock themselves out with a well timed dodge into a wall.
So yeah, forget building a "Batman" talent tree, this is Spidey and I don't even have to do any work! *dusts hands off and sits back triumphantly*