Is a 555/6 suitable for generating a 1-3000Hz square wave from 0-5V control voltage
It feels a little wrong to answer my own question, but just to conclude this, I think it's fair to conclude that a 555 is not really suitable for this specific task.  Thanks to those who commented.

I'm going to go the MCU route - it gives me more control and if I need to broaden the frequency range even further at some point it's then easy-enough to do.