Sometimes you have to play hard ball. Pre-certs and searching for hospitals is not the job of a physician. Talk to your program director and not leave the option. While every residency has some unavoidable scut, what you're describing is somewhat unreasonable.
Social workers don't like doing social work? Too bad. The other choice is, like I said, to admit them, and they get a bill and are transferred in the morning.
I'm not exactly sure how your ER works, but it sounds like you don't have a dedicated CPEP or psych ER. Sounds like you have a regular medical ER that triages psych patients, right?
If so, you're the consult service to the ER. The patient is their responsibility. You can find a hospital for them as a courtesy, but pre-cert and all the other stuff is the ER social worker's responsibility. You don't see them making cardiologists calling hospitals and getting pre-certs for their patients, right? You provide a consult, and the patient is theirs until they are admitted to your service. If they're being transferred, it's not your patient per se, whether they "like psych patients" or not.
Make changes - it's tough but can be done.