I ran into a similar problem, I did have control over all smtps (postfix) and saw the exact same behaviour (only one was active).
I ended up letting the smtp's (postfix) handle queue and send.
I created some DNS A records so DNS would handle the balanced (round robin on each query)
smtp.domain.com A x.x.x.1
smtp.domain.com A x.x.x.2
smtp.domain.com A x.x.x.3
smtp.domain.com A x.x.x.4
and added that FQDN in postfix config
relayhost = smtp.domain.com
You will also need to config each smtp (allowing relay of your server, also I changed queue size and time, etc). They are a lot of info on different ways to do this.
This worked for me.
Hope it helps