I've been in your shoes. I used to work at Banco Itaú in Brazil and spent 6 months trying to get them to let in Apple or Linux machines in a company with like 28,000 windows machines. I also got away with having large plants on my desk even though it was against the rules.

How did I get away with even that much and was it worth it?

To get a new technology introduced, I had to escalate my issue all the way to the CEO of the Bank. This was only possible because I was not an IT employee. I was one of the well paid financial analysts with a fair amount of clout in our area of the bank. I would not have been able to convince the bank of any change had I not worked for a profit center of the bank. As you are in IT, you are part of a cost center (accept this as a fact of life). The only way you are going to make any changes in any reasonable time frame is if someone in a profit center champions your cause. Cost center areas of the bank rarely will be able to raise their issue high enough to make any difference. Banks make money hand over fist so as long as a cost center like IT doesn't get in the way of making money, the top brass just won't care. They won't and no amount of believing that they will will make them think otherwise. Banks are large rent-seeking institutions run by people who just want to make as much money as possible to retire early or keep up with the joneses. I promise you that few if any of these people will care about your plight. They certainly don't see suits and dull dreary offices as part of the problem. Watch American Psycho. Many like the suits (because they are probably wearing Armani and Gucci suits and then spend their evenings in restaurants and bars where women notice that kind of thing). The fact that you are wearing suits will not be seen as problem for them. They may even wonder why you wouldn't want to wear a suit. I was the only financial analyst that didn't leave the bank for lunch without my suit jacket when it was 100 degrees outside and humid. They like the suits. They command power and respect.

Was it worth wasting my time on trying to change the culture at the bank? NO. NO. NO. NO. Don't bother. You'll fight tooth and nail for a year for a tiny concession if you are lucky and you'll always wish things were better. Well they can be better, but not in the bank. Go elsewhere. Go work with real engineers and tech people in a business where tech is a profit center. You will only ever be happy in a company where the job you do is part of a profit center.

Leave. Trust me. Leave.

I've been out of the bank for 3 years now and after working at two tech companies in the meantime, I'm now building my own startup with some friends and learning so much more about tech and business than a lifetime of working at a bank would have taught me. If I had stayed at the bank, I probably would be making a half million to a million dollars per year with bonus by now. I'm currently near broke and bootstrapping and could not be happier. I can't afford vacations or nice cars, but I also don't work a shit job where those are the only things in my life that I have to look forward to.

If you still decide to give it a shot, check out the links from Robin_Message's answer and the book Fearless Change, which is a pattern language book with culture change patterns to help you solve the problems you encounter. I wish I had had that book when I was trying to change things at the bank.