Presumably, an SR-71 successor would use a mix of turbofans and ramjet/scramjet to meet their performance (speed/altitude) goals, depending on whether it would be self-sufficient or use takeoff assist from another aircraft.
As far as the SR-71, its engines were turbojets that transitioned to ramjets above Mach 2ish. They required a special grade of jet fuel and a dedicated fleet of refueling aircraft.
I rather doubt an SR-71 successor would use a turbofan. The mission profile really doesn't require it. I would expect it to be a turbojet that transitions (as you said) to a ramjet and eventually a scramjet.
I would love to see the SR-71 fly again, if only from a nostalgia standpoint, but I really doubt this effort will be successful. I doubt even more that much of value can be accomplished by getting one to be flyable again that couldn't be achieved in a much more cost-effective manner via unmanned aircraft.
Sure, I'll concede here. I think of "exotic" as "few people/nations understand it well enough" but including the manufacture and tooling seems like a no brainer inclusion.