I think, tlloyd, that this ought to be a theory rather than a data post, mainly for the reason rod- mentions: better players are more likely to start in position 2. This means data are most definitely skewed in favor of P2. This doesn't make the data meaningless--but it does make it less meaningful.
Also, we need a starting point, which can only be determined theoretically I think. In the BGG post linked above, the P1 advantage is given as 52-48, but with guided counting ties as "win for P2," which I think is a patently bad way of looking at things; no matter how you slice it, a tie is not the same as a win, even if, as he suggests, a tie is better than a loss. I think Dominiate's method (tie = 0.5 to both players) makes better sense. One could also, I think, argue that ties should be removed from the count when calculating advantages, though the final ratios aren't much different from what I discuss below.
Here are a few of the more standard "basic" strategies, counting ties as 0.5, using Geronimoo's simulator @100k games:
BMU: 55-45
Optimized BMU: 55.5-44.5
BM-Smithy: 55.5-44.5
BM-Envoy: 55.9-44.1
BMU+Colony: 57-43
BM-Laboratory: 56.8-43.2
Using these, P1 should win about 25% more often than P2. (55.5/44.5 = 1.247). This is quite different from the previously mentioned 52-48 split (8.3% advantage to P1). I think this 25% advantage is a reasonable starting point when we consider these sorts of things. Questions like the one posed here are better answered relative to this baseline.
For instance, Militia in Geronimoo's simulator gives a 55-45 split. Militia therefore has no inherent P1 advantage--surprising to me given the strength of Militia.
Double Jack, OTOH, gives a 58.5-41.5 split--that's a 41% advantage to P1 as opposed to 25%. Jack has an inherent P1 advantage.
Mountebank, not suprisingly, gives a 58-42 split, or a 38% advantage for P1.
Witch gives a staggering 59.7-40.3 split, a 48% advantage.
Double Ambassador: 59-41.
Ghost Ship: 56.8-43.2, oddly not that high.
OK, enough simulating for now.