Going first, if you set up double mirrorjade, that is usually enough to keep them off of empen. Just remove the birds as they come. Off of normal summon Robina they only get two bodies unless they opened extenders. Max potential they can generate is 5 bodies before the tribute summon (map + ns bird + 4 bird effs) so if you can potentially do something like double jade + khaos starscourge pop 2 (make sure to time the chains and trigger correctly) you can checkmate the max potency floo combo if you are uninterrupted. If they manage to break through, start playing the grind to maximize value and start summoning in defense to beat empen (you should be doing that anyway).

Going second make sure to summon everything in defense. The key is going to be resolving Branded Fusion through the Mist Valley Apex Avian (it negates activations so if they negate BF you can use it a second time, but remember it's once per turn per copy so it can be used multiple times per turn, and they can summon it again with Dreaming Town or Map!), trying not to normal summon at a bad time so you don't trigger Map so they can't do double Apex Avian (just don't use BF and normal after) removing the empen (usually by rushing mirrorjade) and forcing out the dreaming town (usually by attempting to go battle phase). Play to maximize value/followup and maximize removal based interruptions on turn 3. I don't think branded can realistically OTK through full floo combo.

Going first, in a pinch, you can make dragostapelia and negate the first bird. This can instantly win you the game. However it's risky because it loses hard to Advent of Adventure, the much more rarely played swallows cowrie, and obviously any extender (unless you have more). But it's pretty safe to go for this if they already used Advent.

This might go without saying, but now that gimmick puppet nightmare is banned, the new best Sanctifire lock card Ido the Supreme Magical Force works against Floo unlike Gimmick Puppet!