Except for the "aesthetic" requirement that you note, there is absolutely nothing special about a bass used in church. A four string Precision will work just fine. As in most other music, what you play is far more important than what you play it on. Some of the graphics and inlays that you see on certain metal and signature basses would be unwelcome at church but truth be told, one of our bassists (not me, honest!) does play a bass with a Badass bridge.
As far as the music that is suitable for an electric bass is concerned, any bass that you like and play well is a great choice for church music. I made the recommendations I made above simply because the OP wanted a versatile bass. The only times I have longed for something special in the way of an instrument are times when the music really seems more suited to a bowed instrument, a double bass or EUB. Some congregational music and a lot of choir music would be better served with a bow. But an electric bass will do quite well enough!
Ken