The fact remains that it has been claimed that a 250km distance was traveled in 5 minutes.

That translates to an average rate of travel of 3000 mph, which is a fact.

Given a starting rate of travel of 0 mph and a final rate of travel of 16,038 mph, kindly make that fit.

It would be a fact if you weren't mixing up your units. But OK, I've had a go.

Minute 1 - rocket accelerates from 0-600 - Average speed = 300km/h, distance travelled = 5km

Minute 2 - rocket accelerates from 600-1200 - Average speed = 900km/h, distance travelled = 15km

Minute 3 - rocket accelerates from 1200-2000- Average speed = 1600km/h, distance travelled = 26.66km

Minute 4 - rocket accelerates from 2000-4000- Average speed = 3000km/h, distance travelled = 50km

I'm going to split the last minute into two 30 second blocks as this is where the acceleration is highest

Minute 5 - first 30 seconds. Rocket accelerates from 4000-8000- Average speed = 6000km/h, distance travelled = 50km (because this is 30 seconds, not 60, so divide by 120)

Minute 5 - second 30 seconds. Rocket accelerates from 8000-16000- Average speed = 12000km/h, distance travelled = 100km

Distance travelled = 246.66km. I'd say that's close enough.