Slight tweak to your idea but I can still work it I think.
MC catches ONE of the dogs but Eternatus reveals herself before he can go get the other, the caught Dog and the uncaught one team up with the MC to help take down Eternatus, the uncaught one leaves before the breeding scene and doesn't see it happen.
When the MC goes to get the remaining dog, Sordward and Shieldbert are chasing after her like Eusine was chasing Suicune, thinking that owning her will legitimize their claim as being Galar royalty, they then of course use the particles, the Doggo doesn't Dyna and is enraged instead like you said!