Bottom line is the priest simply has no right to make such a request, especially over something that was a requirement in Canon Law until 1983 (whether practiced or not) and is now optional. The OP can do what she wants in this regard. If he won’t accept a veiled music director, that’s his problem. It doesn’t exactly sound as if the position is no longer available to her if she refuses to not veil and any pastor would be a fool for making an issue out of such a thing. We’re not here to cater to liberal Catholics. They can be the ones to leave the parish if the personal devotions of other parishioners that are traditionally Catholic bother them.
The notion that a choir or music director can wear the attire she pleases is the liberal notion, not the traditional one. I’m pretty sure that pre-Council, a choir director wouldn’t think of wearing other than the prescribed uniform for the choir. It certainly was probably unlikely that the choir director be a woman, except in a woman’s religious community, so she’d be wearing her community’s habit which would include some form of head covering. Or when there was a laywoman leading, all women in the choir would wear a head covering. Nobody would derogate from what was prescribed.
But the notion that someone fulfilling a Liturgical role doesn’t have to conform to the dress code is most certainly disobedient. And it is
disobedience that is the “liberal” view. Obedience is the Traditional one.
In our men’s Gregorian choir, if a chorister doesn’t show up in the prescribed black shirt and black trousers (black sweater allowed in cold weather), he is not allowed to sing with us unless we are singing from a choir loft where we can’t be seen.
The priest most certainly
does have the right to impose a dress code for liturgical roles. The choir director does have the right to wear her head covering… if she is not leading the choir. And the priest has the right to replace her if she won’t conform.
In a perfect world though, it shouldn’t come to a confrontation. Alas in our highly individualistic society, we’re very big on demanding our “rights” but not so much on accepting the responsibilities that come with certain functions such as that of a choir director.
Here’s a suggestion : the OP’s wife can arrive a half hour before Mass (or rehearsal before Mass), put on her veil, and spend that time in adoration before the blessed sacrament.
Then take it off to take up her role in the liturgy. Reverence as she feels it should be shown, responsibilities assumed.