Well I just moved hosts last year. I think today the DNS lag is not as bad. Probably within hours at most for the majority.
But anyway I think I did the same as you are thinking.
Using a dedicated IP at the new site.
I just added a redirect to the New ip at the old site. Closed the board at the old site with a message that stated the move with a temporary IP address URL.
But I recall I had also submitted the new location in my domain registration before I started the move.
I never tried it but it was suggested:
If your new host allows remote access to the MYSQL database. Copy your DB to the new site. Just change the DB paths and login at the old site to the new location. Then if the user is at the old site it will be inserted to the DB at the new site .
