What is the suborbital velocity needed to get from the South Pole to the North pole of the Moon?

The Lunar Prospector impacted the Moon at "about 1.69 km/sec
(3,780 mi/hr) "
http://lunar.arc.nasa.gov/resources/LPBckgrn.pdf

This would be pretty close to the suborbital velocity needed to get from the South Pole to the North pole of the Moon.

Over here:
http://www.freeluna.com/spasnotes.htm
Someone has a "sort of a data dump of ideas that are floating
around in my head."
Which includes section:
USING MASS DRIVERS IN LUNAR LAUNCHES AND LANDINGS

And that's generally the concept I will address with the modification of
max track speed of about 3500 mph [1.6 km/sec] and 2 gee acceleration.

At 2 gees [19.6 m/s/s] it takes about 82 seconds to go from 0 to 3500 mph.
And requires a distance of 65895.2 meters or about 66 km and another
66 km the decelerate, so 132 km [82 miles] of track.

Now making 82 miles of track is no small task, and having Mag Lev get to such high velocities would not be easy. And having two such tracks [N and S pole] is twice the fun.

BUT it would mean that one could travel from the north pole to south pole for "almost" no energy. Granted you of course would lose energy in the energy conversions, I.e there would be energy lost in waste heat and obviously using rockets in "undocking and docking maneuvers" which could amount 100 to 200 m/s in delta-v but I mean that with Mag Lev one could regain energy in regenerative braking, both from "train car" and the payload sent to the other pole.

Since the Moon's diameter is 3476 km. It's circumference is 10,920 km. And 1/2 of that is 5460 km [3392 mi]. It would have travel time just under 1 hr.

And of course this track could be used for other destination other than the other pole. One could "assist launch" rockets going to to Lunar orbit or beyond [L-points, Earth, Mars, etc]. To escape the Moon these rockets would need to add about 1 km/sec. And for rockets landing on the Moon they would also need about 1 km/sec of delta-v.

In addition since they not much energy cost going from pole to pole, payload landing on the Moon could first dock with space tugs at some point during their just under 1 hr transit from one pole to another- so this would mean that the tug would be doing the "aircraft landing on the carrier" maneuver.

So, of course the question is how much does it cost. I have no idea. 100 billion give take a zillion??

So, what could be done in the nearer term is something similar- but doesn't involve 82 mile tracks and Mag Lev traveling at 3500 mph.

You call it a way of recovering the first stage of a chemical lunar rocket.
You simply have the first stage travel for one pole to the other and the second stage going to lunar orbit.