How to pass a parameter to a command that's being executed by exec in bash?

I'm trying to start a subprocess with a unique process name from a bash script. This works:

bash -c "exec -a MyUniqueProcessName './start_service' &"

but my problem is that I want to pass a parameter to start_service. If I do something like

bash -c "exec -a MyUniqueProcessName './start_service param' &"

or

bash -c "exec -a MyUniqueProcessName './start_service $myvar' &"

then it fails with complaining about

/start_service param: No such file or directory

What am I doing wrong?