An addon can do it since apparently, even though equipping individual items is protected, triggering set swaps through the equipment manager isn't. It would be easier if neither set required a talent to equip. I have a DK macro for example that swaps between tank and DPS sets and talents by swapping the set first, then spec.
As you've run into already, this doesn't work for your setup as your enhancement set required the dual wield talent. Some addons provide a delay command or you can use
C_Timer.After() to delay a Lua function call, but this also causes some issues on its own as the timer inherently cannot be canceled. This also doesn't account for any lag you may experience or canceling the talent swap.
If you wish to continue this route, here's an example macro.
Code:
/run SetActiveTalentGroup(1) C_Timer.After(3,function() C_EquipmentSet.UseEquipmentSet(1) end)
This should swap your equipment set 3 seconds after triggering a spec change. I don't remember how long it takes to apply a spec, so this delay may need to be adjusted.