Sorrel Hershberg's reply:
It's tough to maintain your creative integrity, especially early in your career, which I am guessing you are. Forgive me if this is not the case.
If you're working as a freelancer for companies in non-creative industries sectors, it must be hard to make your voice heard. I think though that design teams within other organisations also face similar difficulties in preventing their work being altered by internal clients. This happens to external agencies too, so there is no ideal way to deal with clients who may have a more conservative approach!
You're right that charities and arts organisations can be more open, they also tend to have much smaller budgets at their disposal. If you're happy to work towards a balance of some better paying but less creatively-free work mixed with some more satisying but less well-renumerated projects then this would be an approach to take.
Do some research of charities or causes you support and make a shortlist of those who could do with some design help. Get in touch with them and ask if they'd like to have a conversation about how you could help them. Large charities will more likely already have a relationship with a particular agency, so perhaps go for smaller organisations. Similarly, large arts organisations - like Tate, for example - will have their own in house teams and strict house styles. Smaller galleries might be more flexible.
Also look at the work of some agencies who have a policy to devote a proportion of their time to public sector clients - these are ususally well reported on in Creative Review etc. See if you can speak to their creative directors about what opportunities they have and what you can offer.
I'm not a designer myself, so I'm going to ask some of the other experts if they an also offer some advice.