Hi James,
This is in no-way authoritative but if I had to guess, I'd say they don't have a set formula of spirits for a corpse reviver #2, since it's a classic, it's probably up to the individual bartender to decide. That being said here are my thoughts:
Gin, the ones I see being poured most are Plymouth, Tanqueray, Bombay and Death's Door and Hayman's Old Tom. Of those, I 'like' Death's Door the most, however they all have different flavor profiles that work in different drinks. That being said I'd go with Death's Door, or Hayman's Old Tom. Old Tom gin is a sweetend gin, that's mostly used in classic drinks. Don't be confused with Ransom Old Tom which is more like a genever than an Old Tom. I've seen references to Corpse Revivers with the older styler Old Tom, but if I wanted a more 'modern' gin, I love the soft deliciousness of Death's Door.
Lillet, this one is almost certainly not poured with Lilet Blanc at TVH. I've always seen them sub out Americano Cocchi, which is supposed to be the same flavor profile of what Lilet Blanc was before they reformulated in the mid 80s. It has more of a complex flavor with the addition of the the bitter quinine.
Cointreau, I've rarely ordered anything at TVH with Triple Sec/Curacao, however, the one time I noticed I saw Marie Brizard Triple Sec being used. I've not seen cointreau, but that being said, cointreau is delicious stuff so I think you are fine there.
Lemon juice. I'll leave you to your own devices on this one.
Absinthe, I normally see Herbsaint used in place of absinthe at TVH.
I hope this helps!