By front storage compartments you may well mean the map pocket in each front door panel - the other compartment with a lid on the passenger side is the glove compartment.
If you leave the vehicle with no ventilation then as the car cools the warm air inside will dump the moisture it contains on the inside surfaces. The floor mats and seats will act like a sponge and soak it up. Moisture will condense on the windows and drip into the map recesses in the doors.
When you next use the car the air inside will warm up and absorb some of the moisture - but not necessarily all of it.
And the process repeats - eventually things will go mouldy.
The solution is ventilation - let the car cool down completely before putting it into a garage - leave a gap in each window if you can.
Alternatively water is entering from the outside which it should not do - but ventilation will help to dry it out anyway.
£86 is very reasonable for an inspection - have you told them about the design fault that you've found?- you do not say here.
Also you are not specific about the damage to the car's electrics.
Good Luck...
February 2016