I think it's obvious the Federation would stall out any invasion by the Empire.
Federation vs empire, in a vacuum.....
maybe. In this scenario, no, not likely. Even assuming generally equal firepower ton for ton, the empire just has more tons and more ships, with greater FTL speed, and the federation can't easily devote resources to trying to fight the empire without compromising thier other commitments. They're already skirmishing with the Klingons and need at least some forces there to at least hold the line, and if they move too many ships in they're likely going to have dominion ships come pouring out of the wormhole, so they also need a force there to keep guard. Presuming the Empire is halfway sane and aim for just the federation (or just the cardies) rather than trying to attack the entire quadrant at once, they could easily cause enough damage to force the federation to sue for peace, if not just flat out conquer them.
I disagree, I think the Empire will have issues with power projection. Indeed, the invasion is likely not to be well coordinated because of poor officer training and culture that promotes political social climbers over competent commanders.
That sounds like just a biased sample set, though. Yeah, the empire has a fair number of terrible officers, but they've got a decent number of good ones too. Even in the more cartoonish bits of Disney canon, the average level of capability seemed alright, with a few geniuses and a few fools mixed in.
Saying the empire's officers are poor as a whole just doesn't seem to fit the evidence, any more than saying all the federation admirals are nuts and incompetant just because we see a whole bunch of them that are.
EDIT:
Well, off the top of my head I'd figure that Section 31 and the above board intelligence of the Federation would try to get into contact with the Rebels and other enemies of the Empire.
And....then what? Like, they could maybe help a bit, trade technologies and the like, but they don't have anything that's going to be an outright game changer. Transporters and the like, while useful, are sill blocked by shields and we've known for ages that anything that's important to the empire has a shield in place.
After all getting the Empire into a two-front war would be advantageous to both the Feds and the Rebels et al.
The empire is already engaged in a two front war. Well, one front and one insurgency, but you get the idea.
And the Cardies would be even more willing to play dirty, and so are the Romuluans.
The cardies are too busy getting their butts kicked by the Klingons, and while the Romulans will play dirty, they have no particular reason to do so on behalf of the federation. Remember, when the dominion invaded, they just signed a non-aggression pact and let the feds deal with it themselves, and the Empire is arguably
less malevolent than the dominion.