Hi leaving out other issues for a minute,
your volt drop issue is very valid but not easy, you need to assess total load on whole circuit from house con. unit to garage, work out V-D for that part of run then add in load for skt at end of run and work out V-D for that using both 2.5 & 4.0mm.
? would there be any additional load on garage whilst using Karcher?, unlikely, so what is supply cable to garage, how far from C-U @ 2 kw ie 8.4 Amps approx work out V-D to garage, then work out @ 8.4 A for run from garage to outlet,
total V-D not to exceed 9.2volts
If you state type of cable . length of run method of run ie 'clipped direct' of cable to garage,then can calc.V-D for that part, then assuming SWA for run to skt can work out cable size which 'gut feeling' says at least 4.0mm,
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