Mixing materials is completely fine, however leather is not an acceptable material for a modern kit for actual required plating.
However, I have a suggestion for how to make this work if it is the look that you are going for.
The backplate must go on your vest, then the trench coat goes over the top of that, which means that the backplate can be left off if the trench covers the full back and is not pulled too tight to betray the lack of plating. So you could just go for the basic no backplate because it's hidden by a trenchcoat look... buuut if you wish to attach leather plating I would advise a little more work to make it look like the leather is a fashion statement:
Firstly, I would suggest adding a non leather armour plate to your vest that is just the top portion of the backplate, possibly with a little bit of a neck guard - that would poke out and be visible over the trench coat collar - this would really give the illusion that you have a full backplate made from modern materials under the coat.
You could then add the leather to the coat as fashionable trauma plating (since the actual armour is under the coat already) - to tie this in I would also add some leather panels to the lower half of the coat.
