My digital piano (Roland FP 80) can transpose the existing MIDI file at most five semitones up or at most six semitones down (does not allow to select more). Has it been done for my convenience ("too many features") or there are some fundamental reasons making transposition by arbitrary range more difficult than appears?
Why would you need more than that?
Five semitones up and six down can make every possible key: Let's say the original MIDI file was in C major. Up 5 semitones and you can get C C# D D# E and F. Down 6 semitones and you can get C B Bb A Ab G Gb. I am aware that you cannot transpose to a different octave using that but for practical purpose that is easily sufficient.
I imagine, that that function was designed to allow the pianist to transpose to better meet a singer's vocal range so a total span of 11 semitones is perfect.
Look for a separate Octave Shift function.
As David has already said above, there are two different functions, transpose, which shifts by semi-tones and octave shift which does the same thing by a whole octave. If you just want to shift the pitch by a whole number of octaves you need to use that function, if you then need to adjust by a few tones you need to select transpose as well.
So Audrius needs to transpose down to Fm and then also shift up one octave to get the desired result.