Hello Welles, welcome here.
The compact word is difficult, there are many shades of compact and most of them are just firm, or ‘temporarily squashed’ without being too much damaged.
Unless you have been repeatedly driving heavy loads over the same soil is wet conditions, I reckon soil organisms will do the job for you, with the method you describe.
Not sure you will need the black polythene. No weeds to cover, and best to let the rain in at this time of year.
