The trick to escaping bureaucracy is the same for both. Making friends with people, and talking to them directly when you need to massage the processes. It's just that it is much easier to be friends with 50 people than 5000. Working from home can be an issue, make sure to video chat when you can and goto as many official and unofficial events as you can.
If you have a good manager, and your manager has a good manager, they should have contacts around the company that can help with the red tape. Never abuse it, and when you're in a situation where your urge is to be a roadblock to another team, go out of your way to help them instead. It always comes back in a good way.
Office politics sucks, but it's the only way to stay sane in these environments.
Sometimes you have to remember that the friction is because they just don't know any better, or because they do no better, but they're too busy or too tired to create a better system. They are all dealing with the same nonsense, and it is soul crushing.
In my current job, I had managed to skirt a lot of the corporate nonsense for a long time, but we merged with a company that had a very by-the-book culture, and my manager had to resign for health reasons. Things have been downhill for the past few years, and if I could get a job making the same at a company that didn't do all that, I would probably take it.
..but that brings me to another point, I loved that small company, like family. Then One day a bunch of my friends were fired for personal/political reasons. I left a year or so later, and a year after that they sold the company, and a year after that the new company closed the office and just kept the data center. The point is that bigger companies are more stable, if you have a mortgage or kids it's really no choice.