As a 'box player, I am appalled when I see string players asking a whistle player for a note to tune to! Breath pressure alone can account for quite a difference in tuning, and as Brian says, whistles aren't that accurate in tuning anyway. Not that it matters a great deal, lots of people use the cheaper type of whistles in sessions and it always sounds fine to me. Or maybe that's the Guinness!
I think that using the tuner is putting you off. Were you happy enough before?
Perhaps the answer is to buy a more expensive whistle. There are some great sounding wooden ones on the market. I'm sure that the whistle players'll be along in a minute or two to tell you which of those plays accurately.