Allow only traffic with specific HTTP header string

You could do this in a iRule alone, if you wanted. Maybe something like this:

when HTTP_REQUEST {
     Check if  value matches ""
    if { [HTTP::header ""] ne "" } {
         Doesn't match? Silently discard traffic... Could reject instead
        drop
    }
}

You can use ASM alone to specify that a particular HTTP header is mandatory, but I'm not aware of any way to make the "mandatory" part drill down to the value of the header as well. It's not that ASM is unable to check header values, just not in this particular context. You can have ASM check for the presence of certain headers and header values and block the traffic, for example.