I disagree with the month being wasted. I did exactly what you are suggesting towards the end of my 3rd year. I got a regular letter from the assistant director of the ED who had strong ties to a former residency program. While it is true, it did not carry as much weight as a SLOE from an ACGME PD, people commented on the strength of the letter during interviews.
The month will also help you decide if you are 100% sure you want EM. Basically, my advice is work really hard, impress and try to find a letter writer with name recognition to at least a few local ACGME programs. Of course, switching that rotation to a spot with a program would be better, but if not, you havent wasted a month.