Do you actually want to work there? You are crazy. Here's the thing about negotiations. You can ask for whatever you want. But no one cares about that unless 1) they are convinced you will actually take the job given the right offer and 2) you can justify what you are asking for
How do you justify it?
1. you show them what you are worth by showing them your others offers. Because this is so geographically dependent, they may only be interested in other local offers because you are going to find higher paying gigs in somewhere like california (though probably in the less desirable areas). If others places in the area are paying lower (often academic centers drive down salaries across town for example) they might think the offer is reasonable
2. you need to know what you are worth. This means calculating the wRVUs for 12-14 patients, and also factoring in that they need a psychiatrist to function and generate revenue from facility fees, and possible lab testing, referrals to other specialties etc.
3. You need to find out how much roughly they are paying for telepsych. Remember they will be paying the telepsych company quite a bit more than what the psychiatrist makes. If you are asking for alot more than what they are putting on telepsych, and they have had no negative consequences from using this, they may not be disposed to pay much more for an in-person psychiatrist
4. you need to understand what barriers there might be to a higher salary, and create "pockets" for additional revenue and benefits (for example vacation time, sick leave, paid maternity/paternity leave, productivity bonuses, relocation allowance, CME funds) or negotiating ramps (for example will your salary be fixed, or can you negotiate a ramp so over 3 years your salary goes up by x%
5. in terms of undesirability - that in itself does not transform into a considerable salary bump. Things to consider are the cost of placing job adverts, paying recruiters, interviewing people, going through the whole expensive process, and then the potential of not being able to retain and go through the whole process. You need to be mindful of this and use this as a justification for a pay bump. The issue is the fact that you already moonlight there hurts you, as they are thinking "well he doesn't think it's undesirable if he works here already" so you need to shift from undesirability to thinking in terms of cost savings from recruitment and retention. the other thing is moonlighters get paid more because you are not salaried so no benefits, you pay both contributions of taxes, they are more desperate for coverage. salaried employers will get paid significantly less (though should be in part offset by other things).
6. negotiations are a collaborative process. It's best not to think of it as just a salary thing, as there are often things just as (if not more) important. you need to sketch out your vision of what it is you want. Rather than saying "I want 400k or whatever" (which you can do, but is not going to get you what you want), it is better to not lock yourself in. Present them with your other offers, and the above arguments based on the data you have about how much you are worth and how much better it would be for them to have someone who is going to stay and is a known entity, and have them see what they can do. At that point you can make a counter offer. Now it might be reasonable to make a counter offer at this point but the person with the most power is always the person who is not committing to any particular offer.
7. If you face resistance, ask "help me to understand what the barriers are to this, and what I can do to persuade you of why what I'm asking for makes good financial and practical sense"
As psychiatrists we are lucky in that the job market is pretty good (insofar as there lots of job, most of the actual jobs aren't all that great) so we have more negotiating power than many other specialties. However most physicians are terrible at negotiating, and many of the "rules" of negotiations don't apply to psychiatrists. For example, doctors are often told not to ask for things until later in the negotiation, but for psychiatry I woudn't waste your time with a job you wouldn't be interested if the work wasn't x or the salary isn't y. You can and should absolutely clarify what is important up front (but always be ever so polite about it, especially if it is an person/organization you may wish to work for at some point in the future). Don't be a douchebag just because you can (and for many jobs you probably can).