I think that the most obvious effect that the Mayflower Compact had on the Declaration of Independence was that without the Compact, which acknowledged the sovereignty of the British Crown, there would have been no need to declare independence 150 years later. As far as similarities to the Constitution, I don't see them. The Constitution created a limited government, with no authority over it within lose limitations. The Compact created a government without limitations other than "majority rules", with another unlimited government (the Crown) above it. Near as I can see, they could have just stayed in England and had all that.