I've played the organ and piano for many weddings; however, there is one priest that I have worked with who insists that he should walk in after the bride. This then requires the bride and groom to step aside so that he can pass between them to enter the sanctuary. I've never heard of this and he is the only priest that I've seen do this. Is this normal?
At Catholic weddings in the United States, the priest usually enters through a side entrance and waits at the front with the groom and his attendants for the bridal party to enter. There is no mandated procession lineup though, so a priest could enter first, behind the altar servers, leading the wedding party to the altar (which would, in fact, be a nod to the medieval custom of the couple marrying at the church door and then entering for Mass).
Your priest may be thinking that processional protocol for a regular Mass should be carried over into nuptial Masses as well. At a regular Mass, the priest ordinarily does enter last because he is the presider. At nuptial Mass though, the priest entering the liturgy last is generally impractical for the reason you mention. It’s also not particularly becoming for the priest to insist upon entering last at nuptial Masses, since it makes him look like he is competing with the bride for the congregation’s attention.
The Church usually is quite lenient about accommodating accepted social customs at nuptial Masses, whether that means allowing the couple to kiss, jump over a broom, light a unity candle, or give flowers to the Blessed Virgin. Surely allowing the bridal party to enter after the celebrant could be permitted on the same grounds.