Ok, I'll wait until I get your package to see what's in it. Then I'll call you to get information on antenna length and orientation, what sort of ground do you have on it, what kind of power are you running on it, length and type of feedline, etc.. I suspect if you're running just a 100W radio that a single core using FT240-43 is all you need if your primary interest is the higher frequencies. If your primary interest is 80 and 40m, then I'd recommend a dual-core FT240-43 to get more inductance, but the higher frequencies will suffer a bit in that case and may require use of the tuner.
I've found that the internal tuners included in most radios these days are about useless. Some might be better than others. The one in my IC-7300 is a joke. It will tune the 75m band on my 80m EFHW, but 40m where SWR is 2:1 it won't tune. Have to use the external tuner. My transformer is wound with more inductance on the secondary for 160m where I switch the primary out and switch in a shunt with a relay. This requires an extensive ground radial system to use it on 160 as a Marconi. SWR is <1.1 on the lower end of 160 and rises to 1.6 on the upper end. The IC-7300's tuner will tune that fine, but not much else. It has pretty limited capability as an an antenna tuner.