If it is to be 'counties', then it should read:
"…in THE Napa, Sonoma and Lake counties."
Otherwise – sorry - your client is correct.
Think of it this way:
You could write, “ …in Napa County, Sonoma County, and Lake County.”
or
“…in the counties of Napa, Sonoma, and Lake.”
OR…OR…(and taking the best of both possibilities)
"…in the Napa, Sonoma and Lake counties."
We don’t capitalize ‘counties’ because this is the general term for the plural. Only when we talk about a specific county, and that (for instance, Napa County) is the official name of the county, do we capitalize ‘county’.
HOWEVER………..
You and your client need to look at:
"...the highest tasting room and vineyard…"
Firstly, wouldn't it seem more logical to have 'vineyard' first? A vineyard has a tasting room, rather than the other way round.
How come, in these THREE wine producing counties, there is only ONE vineyard, and ONE tasting room?
See what I’m getting at?
Secondly, whilst we do have the expression, ‘of the highest quality’ and ‘of the highest taste’ (where ‘taste’ means: the ability to discern what is of good quality or of a high aesthetic standard’…
…‘taste’ in your sentence is the literal meaning of the word – to taste food - and we can’t talk about ‘highest tasting’. IN FACT, ‘highest’ takes on the meaning of ‘high’ as: unpleasantly strong-smelling because beginning to go bad. It sounds like we are tasting wines and determining which tasting room serves up the wine that is most 'off' to the nose..
‘best’ would be a better choice, or something starting with ‘most’.
I hope this has given you enough information that you will feel better equipped to help your client make the – dare I say? – right decision.
Feel free to post about this again if you need to make sure you have come up with the perfect sentence.
And welcome to the forum. 
I just couldn’t let it go. I, too, was struggling to come up with the “perfect sentence”.
Let’s thrash this one out:
Experience tasting the finest wines from the vineyards of the Napa, Sonoma and Lake counties."
How many people who visit a vineyard don’t know that it has a tasting room? Those folk who don’t would be more attracted to the ‘experience of tasting the finest wines’ anyway, and then learn that vineyards have tasting rooms; so we get round sooooooooo many problems to do with keeping the sentence succinct.