What you're not seeing is the difference in transmission line length between amp on and off. With it off tune for lowest reflected power on the Icom, with it on tune for lowest on the external meter. If you can't bring it down that way you're doing something wrong with the tuner. As you can see it's a common parallel L series C T section matching network, always use the L tap that gives a reflected power dip with both variable caps. That is to say they must tune within their ranges and not at either end which would put proper tuning out of range. Look inside to see if the knob calibrations conform with minimum and maximum capacitance. If not set them for full mesh, loosen the set screws, set the knobs for zero and tighten the screws. You never know what the blind retarded monkeys did at the factory so it's best to check it out.
One last thing, that circuit leaves a bad taste in my mouth and once a burning sensation on my finger. My MFJ being the same I had to be careful to hold the knobs so my fingers didn't contact the set screws. Doing so adds body capacitance that upsets tuning and being they're hot with RF and could easily be a voltage point an RF burn can result.