alt Hacker Newsit looks like it would be much easier to get a powerful regular fighter jet engine (even old Mig-25 engines were fine for 2.8M), stick it to pilotless frame and just launch. Work a bit on intake shape, modern materials, CFD, and probably you can push it well into 3+.
This is essentially the thought process behind the Hermeus Quarterhorse which is already a thing that exists and is in active development:
https://www.google.com/search?client=firefox-b-d&q=hermeus+q...