There's a whole other, meta possibility which is separate from the "in-story" issues I mentioned in my last answer, so I thought I'd mention it here:
Many high profile celebrities know some form of self defense, so Bruce Wayne having this kind of knowledge wouldn't be seen as particularly "uncommon" these days... even if it was made public.
Question: What do Al Bundy, John Coffey, Rachel Greene, Shallow Hal, Sebastian from Cruel Intentions and Prince Hakeem of Zumunda all have in common... aside from being fictional characters?
Answer: They are all played by actors who have years of genuine martial arts training and experience!
While most folks may not be surprised by some of these actors, others would leave an average person shocked! Personally, I was astounded to find out Ryan Phillipe was a freaking tournament tested black belt in Tae Kwon Do... let alone that Ed Oneil (Al Bundy) only got into BJJ in his 50s and is now a recognized black belt instructor! Micheal Clark Duncan was known for football, yeah....but not being a purple belt in Gracie jujitsu! Jennifer Aniston apparently has been taking Budokan for over a DECADE; she is a very well known performer, but never has she been seen as a "martial arts" star! hell, even Jack Black actually took Kung Fu to prep for Kung Fu Panda... and is apparently quite good! Thing about it, one wouldn't know these celebs were so versed unless you followed them closely, or did the research... and it's rare that you see celebrity trainers "bragging" about their students in real life.
Apparently, though... this isn't an uncommon thing! Many celebrities take classes to avoid issues with dangerous fans, or just for health benefits; others (actors especially) seem get into self defense due to it being part of their characters' profile. For example, Eddie Murphy has been doing action films since the 80s, but got into true self defense some time in the 90s. In an interview, he stated that while having bodyguards is cool, because you are such a public figure, you could be subject to attack at anytime ... so learning some form of self protection is just a sensible precaution that a lot of public figures tend to take at some point.
Considering the amount of attempts on his life, threats to his company, kidnappings and ransoms to his adopted kids, and hell, just his (ahem) "childhood background"... Bruce Wayne would likely be in the same boat. And given his publicly acknowledged "enjoyment of extreme sports", I doubt many would find that too suspect if he added MMA or some even more obscure self defense art to his list of personal fitness hobbies.
While he probably wouldn't "broadcast it", for fear of the comparison, if some famous teacher were to say Bruce had been trained by them in the past, well...
"I've been attacked at my home and offices before"... (which is a documented fact)
"Come on; killer clowns, Man-Bats, Mad Hatters, metahumans and then human gangs, robbers and murderers. I love this city, but Gotham is a dangerous place to live"...
"My parents were killed in front of me by a gun-wielding mugger when I was just a boy..."
"I raise kids here, and I wanted to be prepared to protect them in an
"I've had some crazy threats on my life from ex-girlfriends/stalkers"...
Any of these excuses would make sense as to why Bruce Wayne would take some lessons in how to fight. And considering the popularity of MMA these days...it would actually be more surprising that Bruce is so high profile, and yet hasn't taken such a precaution! However, the whole scenario does present another problem:
Possibly similarities with his alter ego, thus heightened suspicions of being Batman.
This, though, would be a risk more only for his trained enemies,, rather than the general public; for example, the hero Wild Cat was seen as a brutal and very skilled fighter, but even with the same "Signature style," few ever suspected pro-boxer Ted Grant as his identity. Now, any reasonable fighter knows a fighting style is almost as distinguishable as a finger print.... but Bruce already has a few ways around this...
- He is versed enough in several styles and modes, from striking to grappling to military to weapons arts. As such, he does not need to fall into his "vigilante mode" in public, anymore than Saitama needs to resort to his "Serious punch" with most enemies; likely, if threatened, Bruce could do a little MMA, boxing or Krav Maga...all very widely known methods....to defend himself. This way he could stay unharmed, but not "out" himself with Batman-level tactical fighting, master martial artist skill or QCQ.
- The best secrets are those kept in plain sight; Bruce's persona has already been established as a "Billionaire Playboy with No Day Job", as well as the "Altruistic businessman and eccentric adventurer", with both the time and propensity to high adrenaline activities. But that's the brilliance of it: to hide his in-field injuries, true skill level and fitness from training... Bruce has made it publicly known he's into dangerous activities ("cave diving", "base-jumping" and "helo-skiing" in the Nolan films), and often goes on "pleasure trips" that have left him bruised up and injured.
I was looking for a panel to show it, but in one comic, Bruce (as himself) goes to the doctor to have a physical, possible for insurance purposes. Here too, Bruce says he is "an extreme sports junky," and has gotten "a few scraps and marks," as his upper body is a road map of healed scars you'd see more likely see on a soldier or pro fighter. The doctor puts him through a regular stress test to check his blood pressure and heart.... basically has him running on a treadmill and checking his oxygen level. All this, only for the doctor to say, in total confusion "According to this, Mr. Wayne... you have no stress!
Again, this makes sense; Bruce exercises daily, and faces everything from hostile take overs to explosions to gun wielding maniacs to world ending threats from alien Gods; him being "stressed" by a treadmill run would be highly unlikely. But in the comic, he uses "Eastern meditation with a guru, alternative therapies and daily yoga and weights" as a viable excuse...and considering he has pictures of himself actually doing this crazy stuff in different locations around the world, a few on-call personal physicians like Dr. Leslie Thompkins to fix him up whenever, and a public persona as a bit of a risk taker with the means to actually live this life regularly... these explanations aren't exactly out of the question.
I mean honestly, which sounds more likely: Your patient--- one of the wealthiest, most prominent, high-profile, and likely busiest people in the world---- is secretly a highly trained, near-peak conditioned vigilante who spends his nights tracking down and beating hardened criminals into submissions with his bare hands every day .... or that he's simply an eccentric, very active, thrill-seeking, easily-bored health nut celebrity who uses his money to keep himself in great shape with life coaches and holistic therapies, all to indulge in the type of craziness we see eccentric regular people do and document on Instagram all the time?
Oh, and who may happen to have taken some self defense classes from a well known coach in the past....okay, so? The guy is rich, was traumatized as a kid and has a lot of haters... why wouldn't he learn to protect himself in a pinch? That's just logical good sense.