We've sailed at the end of June/beginning of July and really enjoyed it. The cruise ships will bring crowds to Cane Garden Bay on Tortola, the Baths on Virgin Gorda, and White Bay, JVD to name a few places specifically. They are generally gone by mid-late afternoon. White Bay can still be really lovely early or later in the day.
We really loved sailing the Oceanis 41 and friends of ours have sailed the 45 and loved it so I can only imagine the 48 is going to be fun and fast as well. You will likely have a serious draft so if you're planning on Anegada, keep that in mind and check out the anchoring options since the mooring field there will likely be too shallow.
Most closings don't start to happen until August, I believe, so you should be just fine at that time of year.
Don't know much about Dream Yacht Charters as a company but I'm sure someone here can weigh in.