Bike riding around Iceland

Bring a tent. On a bike you have room to pack that. Not only does it make your nights dry, it makes for far easier food preparation, which you can't do properly inside a bivvy bag. If you insist on just rolling out your mat and sleeping bag, at least invest in a waterproof outer cover. But no, bring a tent. There is precious little wood to erect tarp tents or bivvy bag tents in Iceland- which makes wildcamping without a tent far more troublesome than in the other nordic countries.