I have a female name for my surname and I'm often called by my surname because people assume it's my first name.
It doesn't bother me in the slightest.
I have a cousin who has a first name and surname which are both more common as first names. Again, this confuses people.
My granddaughter has a first name which is very old fashioned but is having a resurgence. They then moved house and found out their next door neighbour's dog had the same name.
I think any name has many different associations for others.
I'd go with Cooper if you like it OP.
Nigel Cooper in my class at Junior School had the biggest feet in the whole school. The headteacher used to send for him when the pupils were in trouble. He would use Nigel's plimsolls to beat them with.