I think there wont be a solution. It's possible there's something wrong with the monitoring signal chain but I'd assume the OP has tried different options there to eliminate that problem.
Just in case the three main ones are :
1) Studio Monitors - Signal chain Kemper->Convertor->Monitors
2) Headphones -> Signal chain Kemper->Heaphones (obviously you wouldn't want to use the headphone outs on your convertor when testing in case it's the convertors DAC that's at fault, so pick the headphone out on the front of the Kemper itself)
3) Cab - Signal chain Kemper->Power Amp->Cab, if the Kemper is powered you can just plug right in to a cab with a speaker cable, if not you can always shove the kemper into the loop of your guitar amp, basically using it like a pre-amp. No matter how you do this just remember to disable the cab emulation on the output.
Honestly though, the Kemper really does sound exactly like the source amp 99% of the time. Most guitarists even those that think they really know how amps sound mic'd up tend to bias themselves and imagine that they're more magical than they really are, so cognitive biases kick in.
A couple of things are not captured by the Kemper. One is room reflections. It'll get the frequency response of the room as part of the sound, but not the reverberations or any background noise. You can try to compensate for this a little either with the onboard reverb, or just by dialing up the "space" parameter in the output section. Personally I find the space thing sounds more like recording in a toilet than real room verb, but it's one area where a good Convolution Reverb in your DAW after the event with a nice room's IR can complete the sound.
Another thing it doesn't capture or rather doesn't separate is the microphone. The "Pure Cab" setting (also in the output as well as in each cabinet's settings) is an algorithm designed to reduce the effect of a mic and give you a sound more similar to direct out when using FRFR studio monitors or headphones. It's effectiveness is really governed by which mic was used on the profile, with some profiles it's amazing, with others it can introduce flub.
Then there's the fact that it doesn't capture different settings of the amp, e.g. it would be great if you could take two different profiles, set the Kemper up to know what the settings were on each and then have it extrapolate the inbetween values. Even if it were only changes in the gain dial. As it is instead it captures a single snapshot, and the further you adjust values on the front the further you get from the original amp. It still can sound good, but at the same time it doesn't quite sound or feel right.
And of course if you want perfect separation between amp and cab you will need to do a full separated profile using a DI load box.
Basically the Kemper is truly awesome, yes it actually does sound like the real amps and respond like them too, it remains after 5 years still the pinnacle of amp simulation at this time. But it comes with a bunch of caveats. It's not the end of the line for amp modeling, just another step along the way, a huge step, but still, it's not perfect. It may just not be right for you. You may need to wait till either Kemper or a competitor comes along and overcomes these hurdles.