With fully modulating air intake and f-119, B-1R would be only fractionally slower than the mig-31, but it can stay in afterburner and remain at its top speed far longer than a mig-31 can and thus in reality cover more distance during the time a very long range AAM chasing it can stay in the air, than a MIg-31 can.
Just because you say it can does not mean you can will it into reality. Something to remember is the Mig31 was designed and built from the ground up to spend prolonged periods of time in high supersonic flight, and many lesions learnt from the Mig25 were applied to make it the interceptor it is.
The B1 was only ever designed to make short dashing sprints in supersonic flight and the extreme heat and structural stress such prolonged high supersonic flights would have on the airframe is not likely to be something a few retrofitted mods can adequately address.
In that regards, a B1R would most likely be more like a Mig25 than a Mig31 in that it will have a high theoretical speed, but to reach and sustain that in real life would cause massive and potentially irrevocable damage to the airframe.
Starting from a BVR position at 40,000 feet, 40 miles away, and 60 degrees off nose towards the enemy fighter, a B-1R already at mach 2.3 and able to sustain that speed for 5 minutes will be able to fire its own missiles, turn and out run any known AAM launched from any altitude.
Firstly, where are you pulling these numbers from? On what basis can you claim that the B1R can sustain M2.3 for prolonged periods of time?
Secondly, you seem to have completely dismissed two extremely important factors when coming up with the above scenario.
Factor 1, seeker limitations. You can shoot a BVR missile 40miles away and turn tail, but if the missile seeker cannot get a lock on target at 40miles, you are just shooting a million dollar firework.
Most modern BVR missile seekers would struggle to lock onto a 3m2 RCS target at half that distance, so unless your B1R want to follow its missile in until the seeker has a lock, your KP are going to be terrible no matter how much energy the launch platform imparts on the missile upon release.
To make things worse, the large size of the B1R would allow enemy BVRAAMs to lock in with their own seekers far earlier, thus freeing the launch fighters to break off and start their own evasives, which will further complicate the shot for the B1R since it still needs to keep those fighters locked with its own radar to direct its own missiles.
The second important factor you are not considering is the turn radius of the B1R if it was going to M2.3. I don't think it will be a stretch to suggest the B1R isn't going to be turning on a dime even in subsonic flight. In high supersonic flight, its turn radius and time is going to be massive.
When you are flying that fast, your own inertial is also going to be against you if you want to do a uee, so its going to be even longer before you are doing M2.3 heading away from the incoming missiles.
Add all of that up and suddenly even if you turn tail at 40miles, your odds of outrunning the incoming missiles aren't good. That's just survival. To even think of getting kills, you probably need to close to under 30 or even 20 miles of the enemy fighters, at which point your own chances of getting out of that engagement alive are greatly reduced.
And all of that is just against conventional enemy fighters, against stealths the B1R is just target practice.
Given it's large payload, B-R can saturate the battle space with its own missiles as well as counter measures and greatly add to situational awareness difficulties of the enemy.
Only if the enemy lets you. If the enemy has good situation awareness, they can just divert a pair of fighters to intercept your B1R a hundred miles from any other targets. Feel free to saturate the hell out of those two fighters.
So it's an unorthodox, but far from flimsy, concept. It does depend on BVR missiles being as effective as advertised for the B-1R to be able to shoot anything down. But unless BVR kinematic performance are far more better than advertised, it is safe from being shot down.
As I pointed out already, you have ignored some pretty impact or factors when you were coming up with those numbers, and in a realistic scenario, your B1R is going to have to get a whole lot closer to the enemy if you want to have any chance of scoring any kills.