Mike:
Barker and Flynn may've worked on some of the same courses but I would definitely underscore your paranthetical remark above---eg AT DIFFERENT TIMES.
When William Flynn was basically just getting going in golf course architecture Herbert Barker had left America, never to return again.
It seems to be a bit tricky to pin down in detail what Flynn specifically did on some courses and some projects, in the teens and even on a few projects when he hooked up with Howard Toomey in the early 20s (ie Philmont North). Their company was structured in a rather interesting way for that time----eg a design arm and a construction arm even if it was one and the same company. They also promoted themselves as economic efficiency experts in their course planning and building applications-----viz. it was made very clear to a client what exactly he was paying for in a very itemized way. There is very little question in my mind that this was a dedicated MO that emanated out of Flynn's time and tasks with Wilson at Merion. Wilson as well was virtually fixated on this type of economic efficiency concentration in all things to do with golf----planning, construction and particularly agronomy and its development.
It's a bit mysterious but in the early 20s Alison made some proposal to Flynn that they hook up somehow. It (Alison's "proposal" to Flynn) was only mentioned amongst Piper and Oakley and H. Wilson, so I really don't know what the particulars of it were, even though I can see that ultimately P&O and Wilson did not think it was a particularly good idea for Flynn. It would not surprise me if it was not something that emanated out of the final architectural hole planning and redesigning of those last four hole at Pine Valley around 1921, in which both Alison and Flynn were involved even if perhaps at slightly different times. It may've had as much to do with agronomic development as with design and construction.
Don't forget, at that very time, the Wilson brothers and Toomey and Flynn and Piper and Oakley et al (Marshall, Harban etc) were on the cutting edge of promoting the so-called "vegetative process" (bent) for American golf. That in and of itself was beginning to get a whole lot of attention even prompting Harry Colt who had not even been in America for over eight years to write Wilson asking him to please send over to England all the research material done on this idea.