But hands down, as long as you are getting educated FOR FREE I would personally insist you to take as much advantage of that as possible.

If they pull the I've got an ESA a month into occupancy I will absolutely kick them out not for having an ESA but for not obtaining prior written permission.Having said that you can refuse ESA under FHA there are a number of exceptions such as if causes financial burden on you, if you have less than 3 properties, if you are living in one of the building apartments, I don't have the full list and wording but look it up.I also ask for verification from their doctors, but I can't insist.  

Once, I had an applicant, said he forgot his driver's license, left it home, on the first visit, bought his proposed roommate on the second visit because I insist on seeing people in person renting from me, and left the license home again.

In this situation I would insist on half a month's rent paid on Jan 1 and apply the "last month's rent" to the period Jan 15 - Feb 15.