I don't think the movs can be named as you want. At least, I haven't come across a program to do it yet (and I do it often).
I use Clipwrap to rewrap the mts files or convert to prores. You have to buy it, but it's a solid program and worth the cost (they have plenty of sales on their twitter feed).
But first, I use Adobe Bridge to rename the files sequentially by name using the date and sequential number.
I look at the dates in finder and group the days together as I rename.
My files end up looking like:
Hope that helps.