I would push the due-diligence date back. We just closed on a house this past week. Although did not encounter what you are with this. Actually the sellers offered to be out sooner and would let us start to move in before we closed.
When I sold my house last year, I specifically stated I would not be out before the end of June and would not accept a closing date before late June no matter what. I'd put the house up for sale in April, but had no place to move to until late May. (Hubby was already here, but until I could get a weekend plus a couple of days, couldn't do anything about finding a place to rent. Yes that POed me that I had to find a place to rent from 5 hours away. Thankfully I found an awesome realtor here to help and went back to her when we decided to see about buying a house). The closing date ended up being 2 days after I got everything moved. I am so very grateful to my parents for coming down from Ohio to help. I truly do not know what I would have done without them. They are coming down again for our new house, but not to help move. They are coming to help paint. I MUST repaint the kitchen and laundry room before we move in. They are hot pink. I hate pink to being with and must have those repainted. Also repainting two other rooms. We don't need to move right away as our lease is through the end of May. So we are going to take our time to move.

Sorry the above got so long. I agree with the indysteel. You may not be able to lock in a rate for that long. Now with an inspection, the bank (from my understanding) requires it to be done within a certain amount of time before closing. So if you get an inspection say in April, you'd have to get another one within (I think) 3 weeks of closing.