Supposedly the US did funnel money through the Pope into Poland. I dont know how or even if the CIA was involved.
The whole Banco Ambrosiano is very murky, and the hordes of conspiracy theorists and nuts that have "investigated" the mess have only made things more confusing. If I had to guess, I would say that the Vatican's relationship with Ambrosiano and other similar private banks had to do primarily with avoiding complex Italian tax regulations on the taxation of income received from overseas investment. Its possible the Vatican used funds deposited at Ambrosiano to fund Solidarity, but the relationship went much farther back in time.