They could surely fork it and take it from there. They would rather want that they own the code-base so that their enterprise customers can be assured that it is a product that they control and own.
My product, unintentionally, solves a lot of problems for the enterprise product that my company sells.
Still, in all the discussions here I don't understand what is meant that they "want to own the codebase". This is open source, so everyone who has a copy of the code can do freely within the limits of the license.
So what exactly do they want when they control the codebase? Do they want to make it closed source, or just more input on the development.
Unless they want to close the source, it sounds to me just like a question of negotiation of aligning your and their interestes and it sounds like you could at least partially support your own product on work time.
And of course, you need to balance any theoretical gains by winning a legal argument with your company with your long-term employment interests.
Ah, this is the landmine you've trod on; you can't make a side project that fixes some other product of your company without really annoying the people making the other product.
If they just want the codebase it's a simple OSS fork, why would they need your permission?