と is the quoting particle, でも means "or something" (listing an example just as in
お茶でもいかが？), and the verb is omitted.
In this case it is
思った？, but in other situations it might be
言うつもり？ or something else.
(Did you think) that I would recoil from that kind of anger or something?
The "or something" can have various effects as it does in English, trivializing the idea, emphasizing it, making it sound far-fetched or ridiculous, etc.
As always, there are various ways to translate this and there is no need to be absolutely literal, but とでも is nothing more or less than the sum of its parts.