Thanks a lot guys for your feedback!
I tried putting a pure booster (+3) in front of the amp section and that made the sounds quite a bit better (IMO). Then I performed an experiment where I would record DIs both from Radial J48 DI-box and from my Kemper (it gets its signal from the THRU output of the DI-box). I noticed that the loudness of the DI from the Kemper is somehow ~2db lower (in the screenshot below are the spectra of both DIs – red corresponds to Radial, bright white – difference).
I must admit I'm not super knowledgeable in this topic. Is it supposed to be like this?