Boom did buy a current jet engine for the demonstrator, the GE J85. That engine is used on the T-38/F-5 so there are thousands of them on the market. The demonstrator is actually bad for Boom from a business analysis perspective because it is such an easy project that they have not been able to realize after hundreds of millions of dollars.
The "small supersonic aircraft" which is what the XB-1 is, is a solved problem. "Advanced composites" is irrelevant, any aerospace firm can make a supersonic-capable aircraft out of "advanced composites". Scaled Composites build SpaceShipOne in three years with about a hundred people in a hangar.
But Boom hasn't gotten the XB-1 into the air after five years and tons more cash.
Building a new supersonic airframe: easy
Building a new supersonic jet engine: hard
Literally and actually with no exceptions every single modest-sized aerospace firm can build a supersonic airframe. That problem is solved. Tens, if not hundreds, of thousands of supersonic aircraft have been built and many of them are capable of carrying passengers.
This is not an exaggeration. My employer isn't even in the supersonic aircraft business and we could build and fly a supersonic aircraft if some executive got a wild hair up his butt, in less time and money than Boom. Tiny and embargoed firms in a variety of foreign countries with small or non-existent aerospace industrial bases have done it.
The research into the aerodynamics and stresses on supersonic airframes is done.
Engines are hard. There are only a handful of companies with the expertise and cash needed to develop a supersonic jet engine, especially one that can super cruise.
Boom is saying "yah we're gonna build a supersonic passenger jet, uhhh, don't worry about the engine we'll figure that out later, give cash please".
What Boom is doing is functionally equivalent to an EV startup saying "Yeah we're going to design a car that can go 300mph (easy, individuals {Bob Dauernheim, for one} in garages have done it) that is powered by electric motors (hard) that don't exist yet and that someone else will design" and then getting hundreds of millions of dollars to design the chassis while negotiating with motor and battery suppliers to do the R&D into the hard bit at their own risk with the promise of profits from future sales to all of the people who want a 300mph EV.
My impression is that Boom was not looking to RR for engines for their demonstrator, but for their full-size production model. Those engines would have to be carefully designed & optimized for economic long-haul supersonic operation. (Vs. existing supersonic military jet engines are basically cost-no-object gas-guzzling hot rods.)
Haven't seen any explanation on why Boom can't just buy a few current fast jet engines for their demonstrator.