You seem to think "checks and balances" means "we have so many ways to get what I want through and it's ok to use any of them in any ways as long as at the end what I want is getting done". It's not the case. "Checks and balances" does not mean "if legislature doesn't get us the result I want, we should try courts next". It means each branch of the government has its role, and for the courts it's upholding the existing law, not creating policy. It is clear that the current dispute is a policy/political one - should we follow one policy or another. The courts is exactly the wrong place to decide such things, no matter how many times you say "checks and balances".
> they argue the process by which the rule was passed was in violation of the Administrative Procedures Act
Which may or may not be true (most likely not, since it would be stupid to undermine important policy decision by not signing a proper form, but of course one should never overestimate the federal government workers and their capacity of keeping things orderly). But if it's true, then they can just re-pass the same decision, now signing all the proper forms in the proper places and the end effect would be colossal waste of time and money and the same outcome at the end of it. Since FCC is the one to decide the policy, and head of the FCC has decided to not have NN, arguing essentially "but you didn't fill the TPC report in triplicate!" is not an argument against the policy that can be effective. It's also not a proper way to enact a policy.
> they argue that the rule exceeds the FCC's authority
That sounds ridiculous. Instituting NN does not exceed FCC authority, but reverting to pre-2015 does? I am not a lawyer, but for a common person that makes zero sense.
> by preventing states from enacting their own net neutrality regulation.
Hasn't then 2015 ruling been equally invalid for preventing states from enacting their own policies which do not include net neutrality regulations? If we argue it should be in the state level (which I'm always fine with) then 2015 rule should be rescinded and they should sue for restricting the current rule only to federal policy but allow states to enact their own policies. But don't they sue for reinstating 2015 policy instead?