What difference does it makes that the company is limited liability? No one is likely to experience any liability for their political speech. This is just a group of people using their commonly-owned property and connections to spread their views. Whether the corporation is limited liability is irrelevant.
The fact that you can own and use "stuff" together while not having being liable for the problems said ownership and use might entail is a special exception to the normal state of affairs granted by the society. We make this exception because we deem the benefits outweigh the cost (and there is a cost, make no mistake about it). In order to make this exception we burden the limited partnership with rules and regulations that must be followed in order for us to grant the exception.
It is not therefore unreasonable to say that one such limitation on what the limited partnership can do is promote political speech. We give them extra power for certain purposes, to say that one of those purposes should not be to try and subvert the very mechanism that grants that privilege isn't on the face of it unreasonable.
So, no-one is trying to limit the speech of the partners as such. They have their rights as individuals. And if they want to band together in a partnership to increase their clout, then that's OK to. We tend to historically call such partnerships "political parties". But if they want to use the extra power the we have given them in the form of limited liability partnerships, then it's not unreasonable to say "no". If you want to use a regular partnership, go ahead, but the extra power that goes with a limited partnership is barred from such use. It's not part of the original deal.
So, my point is that people seem to think of limited liability partnerships as fine, normal and dandy, just because they've been around for a couple of generations. They're not. They're a very special exception from how things are normally and how they are "supposed" to be, and that makes them special not only in the extra power that comes with them, but also, not unreasonably, with extra limitations and duties.
They are not "people", and shouldn't be treated as such, and real people can't use them for whatever they fancy. That's not necessarily in the contract.