I think you'll be fine pinching off the buds up until July 4th.
I have A LOT of mums, and mine are budding early too. The mums this year are growing so fast, that new buds are forming as fast as I pinch them off. You also need to remember that from the time you finish pinching your mums (July 4th) until the time they are supposed to bloom is approximately two months. The mums will keep growing and budding during that time, so I really think you will be fine.
Mums are really tough and designed to bloom. They can recover from a lot of abuse and still look great. Just to give you an example of this, my four year old son accidentaly stepped in one of my mums and tore off a whole section of it. No big deal though. It has already grown back part of the way, and I piched it a little bit to make sure it fills out properly. I believe that by the end of August, I won't even be able to tell something happened to it.
Below are a few photos of my mums from last year. This was the first year I planted mums, so they were not as filled in as they are this year. And all of mine were given to me as cuttings, which is why they are all the same color except for one.