Technically VS are not part of Star Alliance, although I understand how you may have been confused by this given your membership of the SQ programme (SQ own a majority share of VS and thus close ties exist between the two).

My interpretation is that you intend to accumulate miles for redemption use specifically on VS - why not sign up with VS? Domestic DL flights in the US will earn mileage on VS (however you will not earn status unless you are flying F). In addition the VS scheme has some pretty generous earning levels for hotels if you go with the Hilton option. Typically stays at Hilton hotels earn 1000 miles per stay minimum as opposed to 500 per stay with US domestic airlines. No affiliation with Carlson though. If you are renting with Avis or Hertz this could also net you 1000 miles per rental. With the 30 weeks planned this could net you 90,000 miles without too much difficulty, thats 2 x economy or 1 Upper Class reward between NY/BOS/IAD and London!!

The program may not suit you long term as a US resident (no credit card available I believe) but may be worth ten minutes looking at the VS website.