Sharing a photo of the dog might be a nice thing to do - especially if your W is missing the dog etc.

However, I think saying you still want to 'work things out' is a big LRT no-no. Your W has made her initial decision to leave and live with her AP. She now needs to face the reality of losing your R completely. She's not going to feel that reality if you are telling her you still want to work things out.

Instead of her seeing you as moving forwards with your life, she will see you as waiting for her...

If you're in any doubt have another look at DR. However, might forthcoming mediation present some opportunities for you to DB?

T 13 M 7
Me 48 H 46
SS 15
BD 7.14 PA
D final 5.16 (H filed)

We receive & we lose, and must try to achieve gratitude & embrace with whole hearts whatever of life that remains after the losses - Dubus