Depending on just how low you cut the established parts the clippings shouldn't bother the new grass coming in.
But to that, why not cut it a little higher so that the new grass isn't cut too soon. You can always re-cut it when the new grass might be cut at the same time.
I am doing exactly the same thing to my front lawn where I removed a dying blue spruce. It was 40 ft high and with it not being there it does change the look...I have to get used to having a lawn there now and I'm waiting for newly seeded grass to come in.