I think that in Typhoon Pact they get cloak from Romes
The Typhon Pact doesn't exist in the STO universe, so no-go there. Furthermore, if the Breen got cloaking tech from the Romulans in Typhon Pact, it probably means they didn't have it before that, therefore they still don't have it in this universe.
But yeah, I wouldn't put it past the Breen to use energy dampening on themselves as an MES of sorts.
I'm not sure this was established canonly, but I thought Breen ships could cloak according to some throwaway dialogue on DS9 and TNG?
It never was at any point.
Not even DS9-era soft-canon material include this.
The explanation in the "Tales of the Dominion War" novel for the successful attack on the Sol System mentions the Breen using a holocloak (which we already have ingame) combined with firing their energy dampener at ESD to disable it.