If birch are anything like maple, once the buds start to open, the sap will taste..well, like leaves. There is also a minor flow in the fall so you might be able to get enough then.
Have been reading a book by Mors Kochanski ( Bush Craft )
This is a wilderness survival book from our friends to the north
of our border, my family's origin Canada!
Well, of all things I find an ancient mead recipe dating from 1717 8)
To every Gallon of Birch-Water put a quart of honey well stilled together, then boil it almost an hour with a few cloves, and a little lemon-pel,keeping it well scum'd; when it is sufficiently boil'd and become cold, add to it three or four spoonfuls of good ale, to make it work, which it will do like new ale; and when the yeast begins to settle, bottle it up as you do other winery liquors, it will in a competent time become a most brisk and spirituous drink, which is a very powerful opener - Moses Cook (1717), The Manner of Raising, Ordering and Improving Forest-Trees.
Hope you have enjoyed it,
Thanks for the link TB!