Your licence will indeed determine your options, if it is from the FAA you'll have to either find an N-reg for rental or fly as a passenger.
BTW if you think of flying from UK to Belgium (or any other Schengen country) remember immigration procedures: GAR on leaving and entering UK, on the Schengen side use only "port of entry" airfields - also but incorrectly called "customs fields". In Belgium this will limit you to the public airports EBBR-EBAW-EBOS-EBKT-EBLG-EBCI.