IcelandAir does weigh your bag if you do not check it. If it is overweight, they will force you to check it. Now, realistically, there should be a little wiggle room here. If you show up with 13.5lbs, they should let you pass since you could just pull out an extra shirt from the bag and put it on to make weight, and that would be silly.
Last year my wife brought along a smallish bookbag (backpack style) as her "hand bag" and I was worried they might object, but they barely even glanced at it. I think anything that will stow under the seat should qualify. Something the size of a hefty laptop bag should be ok. This year I'm going to have a messenger bag with me which is essentially the same thing as a laptop bag.
This is what we do to avoid any checked luggage;
Everyone in the family has a RS convertible bag. In it, we put clothing only, no extra shoes unless they are very light. We pack one week's worth of summer clothing with two warmer outfits. That, plus the weight of the bag, is pretty close to 13lbs. (we also try to wear any particularly heavy items on the plane trip because planes sometime are too cool for sleeping otherwise)
Everyone in the family also has a "hand bag" of some sort with things to keep busy during the trip PLUS anything that you would have packed in your main bag, but would have been too heavy - like liquids, snacks, electronics, chargers, batteries, etc. Oddly, they are not at all concerned about the weight of your hand bag. It could even be heavier that you main bag!
You may not want to lug all those accessories by your feet for the whole flight, so once you are checked in and away from the dreaded scale, you can simply transfer what you don't really want during the flight to your main bag & then you are set to go.
Interestingly, Our bags were not weighed at all on any of our return flights - only outbound - but I would not assume that was always the case.
