Don't fully blast the whole time. Just go horizontal and turn off "extra lift", burst till you get to where hydros no longer work properly (at around 7km altitude) and then go vertical and pulse your hydrogen thursters until you're out of the gravity well. (fire till you get to max speed and then cut thrust until you're to half speed, then repeat)

I built this thing to go up to space, down to the surface, and up again on one tank and could do it with some fuel left to spare for docking. This model's weight is increased but it shouldn't be enough to degrade that performance by much. At most the missile could be weighing it down, but the internals shouldn't be much heavier than the base model.

I'm using a lot of hydrogen to get out of the atmosphere, what should I do?