Modern thinking is not to do anything to a cut, the tree will heal itself quicker that way. Sealing it also is supposed to 'lock' in any pathogens which might already be present.
As long as the tree is not weeping sap from the pruning cuts, I would tend to leave it alone.

I suggest to do it in the dormant season, well before spring, when the sap is falling rather than rising

Definitely do it before the sap starts rising. Birch have this nasty habit of bleeding sap if done at the wrong time. Sorry, should have said that before.
Could always make wine out of the sap though, supposed to be very good.

So could I prune now? Or better wait until leaves have fallen which I suppose the dormant season?

No don't prune until the leaves have fallen, they fall as the sap falls in the tree.
In the spring next year it will rise again pushed from the roots so it must be before then.
usually it between november to march but you hsve to watch the tree,
this year spring was early so march would have been too late

so look to November time.......

Thanks PamG. Will do it sometime in November.