Volume,
A good way to assess relative induced drag is to consider the far field.
Plot the total configuration lift for all lifting surfaces in units of force across the span of the configuration. Lift at the tips should be near zero.
Minimum induced drag for a given span implies a smooth distribution with the minimum slope at any given spanwise location, with an elliptical loading the classic "ideal".
As we both agree, the canard has to be highly loaded, so it tends to cause a lump of lift in the middle of the wing, with a high slope discontinuity at its tips.
You can reduce this slope break by unloading the inboard wing and the canard downwash will help you do it.
But this means you waste the inboard wing area as it loafs along, not carrying much lift. Wasted area means unnecessary skin friction drag.
A section of inboard wing with inverse taper might help you out, but that induces structural problems since you probably still need to put the gear in the wing.
I glad you mentioned the need for canard high lift devices and their complexity.
Unless the wing is over sized (more wetted area), it probably needs its own high lift devices, simple trailing edge flaps and maybe outboard slats.
In any case, you now have high lift devices on two lifting structures to maintain as opposed to the conventional configuration needing them on only one.
Hardly a way to achieve greater efficiency.
I'm not really anti-canard, but I've been through this trade a couple of times in my career. Canards are great if you want an airplane with gentle stall and a limited CG range.
They can also are great de-stabilizers for fighter maneuverability.
Canard configurations don't work well for airliners where you need a highly efficient cruise capability, relatively simple high lift devices and the ability to have a wide CG range for loading flexibility.