Right. I think you phrased it just fine. The brushes kit does not have the exact same sounds assigned to the various midi slots that would be found in a default General Midi drumkit.
Here is the General midi drum map:
When I import this mid file and open it in Log it appears as follows:
The midi notes are properly assigned to C1, kick; D1, snare; and G#1, closed hat.
The notes should not change from one DAW to another. The other possibilites are that your track in Bandlab it set to transpose by, perhaps an octave, or perhaps several semi-tones. This would cause the the “wrong” notes to play. As long as your drum kit adheres to General Midi standards, you should be getting a Kick, snare and closed hat. Are your midi notes in Bandlab at the correct midi values. If the show numerically rather than by note, it would be 36, 38 and 44?