In the context of his own universe, he does. Nolan Batman is implied to be on the level of DCEU Batman or Comic Batman, it just isn't backed up by feats, which is the only thing that counts here, granted, but narratively, Nolanverse Batman is an elite fighter. It's the same logic with Buffy. She's theoretically supposed to be on Captain America's level of fighting skill and strength. She isn't, but storywise she's supposed to be. That being said, he's decent at prep, and I could see him John McClaning his way against more than a few MCU people(Well, a couple).
In a H2H fight or random encounter where both are equipped with gear (barring Baleman is sitting in his Tumbler or Batwing), Punisher beats the pants off Baleman. He had a couple stalemates with Daredevil and fought - with DD's help - out of that Irish mob's dungeon after being beaten down and a hole drilled through his left foot.
Under these stipulations, which admittedly massively handicap Frank, I agree with Dr. Will Hatch that Baleman has enough prepping (the skyscraper in Hong Kong sequence in TDK) and stealth (docks in BB, skyscraper sequence at the finale of TDK, tunnels in TDKR before the fight with Bane) to take down Frank here.