I told Trafton and company about this because I'm working on-site at WPEngine this week and this looked very relevant to their interests. They specifically want to rescue your site (gratis) so that they can use it as an example of "Wordpress hosting is hard to do right. You don't want it to blow up on launch day. We're very experienced with not blowing up on launch day."

i.e. He doesn't want to sell you on WPEngine, he wants to save your bacon and use the bacon-saving to sell other people on WPEngine ;)

Jason Cohen (their CEO) says "We'd handle the migration immediately and give him a year of free hosting to get a case study out of this."