Jim,
Yes, it sounds like Barker was involved in the construction process, and particularly the process of constructing bunkers, much as he had done at Garden City. He left Garden City and became pro at Rumson in April, 1911.
The strange thing is that Travis's magazine, American Golfer, reported in May 1909 that Barker had laid out a fine course at Rumson. Frankly, I think Travis was trying to help his friend Barker get started in the design field at the time and likely had something to do with Barker getting the pro job there.
Patrick,
Yes, Barker was commissioned to design Arcola CC in September, 1909 at the rate of $25 a day. The course had a soft opening in October 1911 with a member's tournament before opening for real May 4th, 1912.