I never found any reference in Ross' extensive private letters, telegrams, whatever, to anything doing with Andrew Carnegie.
Ross left Dornoch in mid-March 1899 for the States, returned for the first time Nov. 1904, stayed a month to get married, left in December, returned in August 1910, then not again until after WWI.
John Sutherland at Dornoch despised Ross. The only design tinkering Ross ever did there was on the 2nd green and Sutherland regretted it. He made no mention of Ross in his famous 50-year anniversary speech as club secretary (1933), wherein he extolled all the great names associated with Dornoch and never mentioned Ross in his 30-minute speech.
Ross was ignored as a son of Dornoch until a plaque finally went up outside the cottage in town where he lived, but it was not placed on the house until around 2001.
That doesn't mean they never met or even worked together. It just means it's less likely than a casual view of their lives and overlaps might suggest.