I’m very sorry for your loss. Having had a beloved pet that also had to be put to sleep, I know how difficult a pet’s death can be.
Unlike humans, whose souls are spirit and thus survive death, animal souls are material. That means that animals cease to exist when they die. This does not mean, however, that God cannot recreate the animal souls. Perhaps, at the end of time, when the heavens and the earth are made new (CCC 1046-1047), God will create anew the animals who were our beloved companions in this life, not because he has to but as a gift to us.
Because animals do not have spiritual souls, they do not enjoy the Beatific Vision, but, perhaps, because Christ is both God and man, they may be able to interact with God made man even though they do not share in the spiritual joy reserved for spiritual creatures such as angels and saints.
All of this is speculation, of course, but it is something for which we may hope.