Lucky you! Ever since I first visited Iceland I've wanted to cycle there, but I haven't had the opportunity. A few points.
1. Iceland is experiencing a tourism boom. Consequently accommodation is in short supply. I would want to have a suitable backup, i.e. take a tent.
2. About half the population (which isn't very large) live in Reykjavik or its environs. All the population live close to the coast. Central Iceland is uninhabited, the roads (in the widest sense of the term) across it are only open in summer and then only to 4WDs. I'm not sure if they will be open yet. The point is, once away from Reykjavik treat it as expedition touring and go prepared.
3. As suggested above, I would take a tent. The bare minimum of equipment to go with it would be a sleeping bag, karrimat, stove and cooking gear. And some iodine tabs, you may end up drinking out of streams and whilst they ought to be clean, best to be safe.
For option A, the worst that could happen would be hypothermia.
Given the opportunity to cycle in Iceland, I would go straight away. So I would take your option B. Though if you want to delay, fair enough. It really depends how experienced and confident you are with cycling in remote places.