The biggest question I would have is this: for Game 1, if FD is chosen after striking out the rest of the stages, who gets to decide which, if any, Omega stage is played instead? Whoever struck last already got to select FD. Should he/she also get the opportunity to pick an Omega stage that would give him/her another leg up on the opponent? However, if we let the person who did not strike last get to select a particular Omega, would that stage swing the balance enough so as to be described as "punishing" the other person for picking FD? Others like SmashCapps know better than me exactly how much the different properties of the different Omega stages change the gameplay compared to that on FD. If it is that significant, then I would advocate having Omega stages not be available for Game 1, unless both players agree to a specific stage (one use of a general "gentleman's clause").
My rule for the remaining games in the set would be simpler. Because only one person (the loser of the previous match) has any involvement with selecting a stage for Games 2 and onward (by contrast, both are involved for selecting Game 1's stage), that person should have the freedom to select which Omega stage, if any, they play on if that person decides to counterpick FD. Keep in mind my philosophy also includes counting FD and Omega stages as one entity during striking/banning, with FD as the "main" one - if FD is stricken/banned, then the Omegas are stricken/banned as well.
The main problem that I see with this is that there are many more Omega stages available than regular stages that will ever see the light of day in tournaments. Say you're playing against Ness (as someone other than Ness, mind you
), and you want to ban Omega Onett so your opponent can't use the walls to help recover. However, regular Onett will never be used in a regular tourney ruleset anyway. Using one of your stage bans on an already banned stage just so you can prevent the opponent from using its Omega form doesn't seem like the smartest use for them.