I don't think this makes the Federation look weak.
It does continue to paint the Republic in a bad light though.
The reason I think it makes the Federation look weak is because it sets the precedent that they don't respect their own sovereignty. It's also completely out of character for them given past reactions to invasions. Caldari invade: fight a capsuleer proxy war with them. Amarrian House invades: invade right the hell back. Republic invades: Oh my gosh I'm so sorry I made you angry here I baked you a cake please don't be mad at me.
If anything, if an ally backstabs you, you need to respond MORE firmly than if an enemy attacks. What the Federation has done has sent the message to the Republic that it can basically do whatever it wants to the Federation and get away with it. I mean, shit, where do you go from
invasion? Actually taking and holding Federation territory?
Things the Federation could have done differently:
Met the invading fleet with overwhelming force. Don't just matching their dread count, drop 50 dreads with carrier and supercapital support. The idea isn't to win the fight, but to make fighting an unattractive option in the first place. (I'm making the assumption that the Federation can activate a cynojammer once their fleet jumps in so the Republic can't escalate.) If no shots are fired, great. The situation is defused. If the enemy DO fire shots, the battle is over quickly and with a minimum of casualties on your side. This is important in a society like the Federation; the families of thousands of slain crewman can (and certainly should be) go on a media campaign to raise awareness for the slain and demand that the Republic be brought to heel. Minimize casualties, minimize negative public opinion.
Not given Broteau without an apology. You have something the Republic wants and they JUST put several thousand of your crewmen to the sword. Why the hell are you giving them shit for free?
Made more noise when Broteau was executed. Ok, you wanted to give them rope to hang themselves, and they did. Why are you letting them off the hook?
I'm not a big fan of the whole "warmongering facists" flavor of the Federation, but neither do I find "bunch of wimps who can't stand up for themselves" all that interesting, either.
And yeah, some more info on the Republic's reasoning throughout this process would be nice too. What political reason do they have to be so belligerent, besides the fact that they can apparently get away with it?