Left luggage
If you do the LHR-OSL-BGO trip, you probably will have to pick up your luggage and recheck it at OSL in any case (LHR is AFAIK not one of the origins that allow direct transit of INT-DOM luggage yet).
If Park Inn doesn't want to hold your luggage (or you don't want to make the 5 min walk each way), OSL has a manned left luggage office at the arrivals floor. Turn left after exiting arrivals, it is at the end closest to Park Inn (i.e. the opposite end from the train station). The cost for a suitcase is NOK 101 for 24 hours, or NOK 326 for a week.