You could get your builder to do a "turn key" build for you. Basically the builder buys the lot of your choice and builds your custom house on it. Then he sells you the whole thing. No construction loan needed since your builder owns the property while it's being built.
This. And if you own a lot already, I believe the builder can buy it from you and still do the turnkey deal.
To answer the OP, you just need to have a case number before the deadline. You can close after that date and be grandfathered in.