"THought experiment: how much lower would the delta-v to orbit have to be at the south pole in order to make up for all the added hassles of launching from the middle of Antarctica?"

Good question, but it more than simply a logistical problem. Political aspect would a significant problem also.
But disregarding the other problems one could look at Sea Launch as a real example of a launch site which deals with the logistical problems precisely to lower the Delta-v to Geostationary orbit. Sea Launch is saving somewhere between 10-20% in the delta-v cost to GEO by launching at the equator. Their delta-v saving to LEO are relatively insignificant- somewhere around 1% at best.
Sea Launch spent about 2 billion dollars for all it's launch
infrastructure. So obviously, if the Antarctic launch infrastrcture were to cost around 2 billion and it was lower the delta-v cost to all orbit by around 20%, then in rough terms that could be viable.
That leads to the question how much would this launch infrastructure in Antarctic cost? Well, to add a launch platform at KSC for Delta-IV type launchers costs around 200 million. And probably the launch infrastructure for the Shuttle could easily exceeded a billion dollars- and that is using the transportation infrastructure which pre-exists- roads, rail, ship [barge] not to mention the most important aspect having an available workforce in the area.
For the govt to put the launch infrastructure for something like the Shuttle in the Antarctic, the costs could easily exceed 100 billion- and even if it could cost less than 1 billion dollars their would little political will to make it happen. Only non-govt launch site are near the Equator [Ariannespace is quasi-private- I would say more private sector than say Airbus].

On the Moon one would have a lower delta-v to GEO, all earth escape trajectories, or Earth L-points at lower delta-v cost of around 70-80% and about 50% less to LEO. On the Moon a sub-orbital sounding rocket or ICBM is an orbital and/or escape rocket.
SpaceShipOne has one stage which is an air breather, but disregarding that aspect, it's delta-v on the Moon would get it to Mars, GEO, the L-points, and possibly to LEO.
Or look at the Apollo LEM- it went from Lunar orbit to Lunar surface and returned to Lunar orbit. The amount Delta-v it had would not allow it from an Earth which had no atmosphere to reach Earth orbit [or reach the surface {safely:} from Earth orbit].