How much of a patient fee and shift fee should I ask for if I plan to only work one Saturday a month.
Think about how long you reasonably expect it would take you to perform whatever task it is your expected to perform. If it's a fairly straightforward "staff this case with me" request, my guess is that would likely take no more than 15-20 minutes, likely less, though, of course, some cases may take much longer. Assume you are working constantly, so that means you would staff 3-4 patients per hour. You know your base rate - $50/hour - so base your "per-head" fee on whatever you think is a reasonable hourly rate, again assuming you're working for an entire hour. Depending on where you are geographically and whatever "the market" is in the area around you, I don't think aiming for an overall rate of $100-150/hour is unreasonable. That's what I'm getting paid at my gig, and this is in a place that doesn't have an extreme shortage of psychiatrists. Most other gigs in this area pay in a similar range.
For what it's worth, a couple of co-residents were looking at a moonlighting gig for inpatient rounding over the weekend, and they were being offered $25/patient. That seems unreasonable to me since that includes 1) reviewing the patient's chart, 2) actually seeing the patient, and 3) writing documentation, all of which would likely take at least 15-20 minutes in a best-case scenario unless the case is super straightforward and/or your documentation is extremely brief. If a patient monopolizes your time and/or some kind of complication develops that you have to manage, poof, you're now getting paid $50/hour or less, which isn't worth it in my mind.
Ultimately I think you should assume the worst and negotiate your fee based on that. The other thing is that the worst that can happen is that they will say no and you'll be in the same position you started in. I agree with the above, don't undervalue yourself. If needed, perhaps change the per-hour/per-head fee around. The hospital isn't going to want to pay you all that much to simply be on-call and sit around, so offering to reduce your hourly rate in return for a reasonable per-head fee may be more likely to move the needle.