An old unanswered but very interesting question.
On trusting the bartender, there might be an analogy with the kashrut of olive oil. R Yirmiyohu Kaganoff writes here that the OU holds that extra virgin oil is reliably kosher regardless of its source and without any other indication. Their reasoning is that although there is a long litany of adulterations and fraud, all have been with vegetable oils and other vegetable sources and none with overtly non-kosher products.
Applying this analogy to alcohol would mean that
- for those following poskim ruling that some alcohols are kosher by default (e.g., unflavored bourbon, scotch not fermented in wine vats), these would remain kosher even if opened because there is no economic interest for the bartender to mix in non-kosher spirits
- more expensive alcohols (e.g., expensive single malts) which might be doctored by mixing cheap spirits (e.g., blended whisky + sherry, sake and wine) would be suspicious
I checked with a Rav regarding other issues mentioned by @Daniel (e.g., spills from one bottle to another, bottle touching food) and he didn't find them troublesome, either because they are very low-probability events [see here] or because the bottle and glass are cold . Two important assumptions is that the glass needs to be clean and that no fruit (e.g., lime) is served with the drinks.
For further sources, see here. On the appropriateness of drinking with non-Jews see here, here and there.
Of course CYLOR before applying this in real-life.