Frode Solheim assisted me on the FS-UAE User Manual by providing the following solution - mapping an unused key as the right mouse button. This allows you to press the actual left mouse button on the Mighty Mouse and an assigned key simultaneously on the keyboard to simulate the special click.
This is done by setting the following option in the configuration file:
keyboard_key_tab = action_joy_0_2nd_buttonNote that this is global, so if you require the tab key to be usable in other applications or games, comment the line out before starting FS-UAE.
My thanks go out to Frode for his help with this - it served to allow be to play the game properly before my replacement Logitech M500 mouse arrived.
By the way, for an older game, The Settlers works very well with a well-specced emulated Amiga. Mine is defaulted to an A4000/040 with a lot of fast RAM, and the game happily uses this for additional effects.