I would have to agree with the group here. Although I have gotten flamed for it, I dove the Blue Hole right after my OW certification. They will take you, they say it's fine. They do it all the time. It was a very professional operation and you can tell the guys have done this over and over again. BUT, as the doctor mentioned you don't have enough experience to know how you will act at 135'. You have nitrogen narcosis to contend with and more importantly (IMO) is your air consumption. As a brand new diver your air consumption is probably on the high side and the anxiety you will feel at that depth in an overhead environment (you do a little swim through the stalagtites and such) will probably make it even worse. I was pushing it on air on the way up and there was little room for contingency.
I will say I thought the day was very rewarding. The other two dives you do out there are amazing and the Blue Hole is pretty neat also, especially the sharks at the top. In retrospect, however, I probably should not have done it so soon. I think my skills as a diver are much more developed now and I would put a lot more planning into that dive now than I knew to do at the time. Just my $.02 from someone who did it in your situation. There are dozens of good dive sites much closer to shore (5 mins vs. 2.5 hours) and will offer just as much to see. Good luck with your trip!