Take the Airport Express by Grayline - https://airportexpress.is/
And select the stop "T10 Hotel, Nordurey Hotel Reykjavik Road, Reykjavíkurvegur 72" - you'll see on a google map that it's quite near The Flatahraun place, like a 15-minute walk. Maybe they let you store the suitcases for a few minutes at that hotel while you walk to the camper rental and pick up the vehicle.
So that's in Hafnarfjörður, for your information. It's part of the capital area, but technically speaking it is not Reykjavík.
PS: did you say "cheaper than 320 euros"?
Edited: 5 years ago