It is the pastor’s right to set the dress code. Many head coverings (of any sort) and many hairdos are distracting. They draw attention to that particular person when attention should be elsewhere. If this priest sees that as a problem or even a potential problem, he has the right to “correct” this issue.
Also, coming from an ethnic tradition where many still cover their heads, the idea of a veil would be distracting. On my hisband’s side the women wear rebozos which they pull over their heads when they enter church and slide back off when they exit the church. All of the women completely remove their rebozos if they are performing a ministry at the church or during Mass. They don’t wear them at all. Not so much for uniformity but because it gets in the way. Having to readjust and such is bothersome. On my side, the women wear hats (myself included). Do I feel God calling me to do that? Not really. It’s just my family tradition. If I am wearing something and don’t have a matching hat, i go without one. Otherwise it would be distracting and look silly. If the wife feels it is so important to wear a head covering, a veil is the least traditional and most cumbersome to choose. She may find it is much better tonresearch her family tradition, make adjustments as needed and then speak to the pastor. If he still feels it is too much, cover the way my mom did is when we were too young to wear hats without fidgeting. She would use a headband or even a large hair clip. It covers as much as many chapel veils and they don’t fall off as easily. Another thing she would do is tie our hair into a bun or a twist and cover it with a mesh bun cover. This is more how the “old tradition of headcovering” was done than the frilly veils.
