I would pick first week of June.
We normally go the 2nd week of June, and the crowds seem to be more manageable. We are in Orlando every 3 yrs during this week, and the weather is anything from unbearably sticky (a few weeks ago) to cool and rainy (6 yrs ago - Universal was dead empty that week). We like this week bc it gives us time for school to be extended for snow days, but it's before most of the NE schools let out. I was still seeing graduation photos from my friends' kids in NY & VA on the drive back to KY.
This year in the second week of June, we managed to get 4th FPs most days we wanted them and the non-FP rides had moderate waits for the most part. I don't think we waited more than 40 min for anything, and it was much less for a lot of the rides. We did not get to ride some of the E-ticket rides (no FOP, no 7DMT) that we were interested in bc we were offsite and just could not get FPs and did not want to wait, but we had a great time and rode absolutely everything else we wanted to ride with no real difficulties. Now, there were kiosks and restaurants that were not open bc it was not peak, but our waits at most places were minimal, and we got same day ressies at times we would normall eat at both Spice Road Table and Rose and Crown.