I am currently in the UTPNCM program and I am not too sure on the letter of eligibility but you will eventually need a letter of unconditional acceptance. This is my take on the situation:
When applying for the UTPNCM program, you just need to have your application into the universities and then you can provide "proof of application". Your UTPNCM application will go through the process and then in the spring you will get either a letter of selection or non-selection. There are lots of members who are taking university courses in the fall semester when they are applying to UTPNCM program in order to meet the requirements. Marks do not come in until after Christmas and then they are forwarded to the university that you applied to for consideration. That was the case for me.
By the sounds of it, you have already met all of the other requirements for the UTPNCM program and all you have to do is get your application into the universities, provide your proof of application, wait for your university application to be processed and then get the results to the BPSO and DMCPG 5-2 by 1 Jul 15. I recommend getting the letter in as soon as possible.
4) ... IF LETTERS OF UNCONDITIONAL UNIVERSITY ACCEPTANCE ARE NOT AVAILABLE AT THE TIME OF APPLICATION, THEY MUST BE SENT DIRECTLY TO DMCPG 5-2 NO LATER THAN 1 JUL 15
So as you see, you will need a letter of unconditional acceptance at the end of it.
If the university is not taking applications for the fall, apply for the winter semester and then have it deferred. If you are a keener, you could submit an ILP and take a course in the winter to get a jump start on your degree. This could work to your advantage if you are not selected this year as it will show initiative and academic aptitude on your part. If you don't want to or cannot take courses in the winter, you can always defer your application.
So to answer the OP's original question: what is a letter of eligibility? No idea but it may be easier to get the proof of application.
As far as deadlines go... They aren't too bad to meet as long as you do not decide to apply at the end of the summer. This time of year is pretty crazy at the universities.