Sounds like a similar situation my girlfriend was in. Other relationship lasted 6 years and for 2 years after it was shared custody with the dog (which ended not that long ago). The only nice thing about the shared custody was when the pup tore her ccl and had to have tplo surgery. Had we done it locally, it would have been over $5,000. Instead, she drove to Washington state university where they have a vet program. We got the surgery done for $3,000, of which half was paid for by her ex (the only time I was happy about sharing the dog with him).
We bought a house together a year and a half ago and I have always wondered what would happen if we split. Sounds like you are living the nightmare I fear. Sorry to hear, bud.