Well, in theory, a Centaur starting from about 30,000 ft and Mach 1.7 might need about 8600 m/s to hit orbit.
With these numbers from Astronautix (I know, I know, always a rsiky beginning to sentence but hing with me), Centaur has a dry mass of 2,026kg, and 20,800 kg of fuel with ISp 451s. So the mass ratio for 8600 m/s can be:
The burnout mass is equal to fuel divided by (R-1):
Of this mass, 2026 is the Centaur stage itself, so the amount remaining for payload is about 1470 kg. The first stage would need to be roughly two F135s, given a thrust under afterburners of 191 kN each and a total stack mass of anything between 16 and 32 tons. Given the Centaur and payload will mass about 24.3 tons, I think that's a safe estimate.
Your estimate has one big fault:
Centaur lacks the thrust it needs to accelerate itself when staging so low.
RL-10 has thrust of 99 kN. This means it can lift 10 tons against earth gravity. But it weighs 23t so it has T/W ratio of under 0.45
The jet-first stage can only put the craft into ballistic trajectory that has apoapsis at 21 km. So the centaur would have to contribute quite big part of the upwards acceleration (which is cannot do very well because of the bad T/W)
This rise on free ballistic trajectory would take about 48 seconds.
But if the centaur is firing downwards(at T/W of 0.5) it would rise up to 33km,
and rise would take 96 seconds. (of course the T/W changes slightly , but even after ~100 seconds the centaur would still be way below T/W of 1).
So, we get a nice suborbital flight that goes only to 33 km. Then we start falling, with a fuel tank that has lots of fuel left.
Just before we hit the ground we might reach T/W of over 1 and start slowing down our fall.
So practically 2-engined, 170kg heavier centaur is needed. And it still won't have T/W of 1 on staging. But as the T/W will be close to 1, it can consume enough fuel to get T/W over 1 before the apoapsis, so can actually really accelerate itself upwards. And after the upwards acceleration, also horizontally, to get orbit speed.
But I claim it will need more than the 8600 m/s delta-v to reach orbit due the terrible gravity losses, and as it will be 170 kg heavier, payload will be smaller too.
8.9 km/s delta-v and 2 RL-10's would mean payload of about 1030 kg,
but I think even the 8.9 km/s is too optimistic.
Centaur really needs a first stage that can stage at much higher speed and/or much higher.