Your planned itinerary is too ambitious. You might make it, but might is not enough. Considering the uncertainty of hiking off-trail in unknown terrain, you need to have a plan that allows shortening/escaping if your route fails.
I did my first trek in Iceland last year, in Lónsöræfi. I have previous hiking experience in the Alps, Swedish Lapland (Sarek, Abisko-Kebnekaise, Nordkalottleden), northern Norway (Rago, Hinnøya, Senja, Sørøya, Narvik mountains), Canadian Rockies (Jasper National Park). My most difficult hike was in Iceland. A trail on an Icelandic topographic map is not always a trail; it might just be a route, which might mean no more than a cairn every couple of km. A route may cross deep, wild rivers and may require scrambling along steep scree with raging rivers below. During a one-week hike, I saw zero other humans in a week of hiking. Such a lonesome hike I've rarely seen even even in remote off-trail hiking in Swedish Lapland.
Considering that Iceland is very different from the Alps or Scandinavia and you are not experienced with it, you should choose an itinerary that is less ambitious and more flexible. I expect some parts of Iceland are easier than Lónsöræfi, and if you hike on fjallvegur you can make more distance, of course. So maybe you can do it, but maybe is not good enough. Choose an itinerary you can shorten or lengthen based on your progress, get familiar with the landscape, and then decide if you can attempt the coast-to-coast hike another year.
Hnappadalsá in Lónsöræfi, 20 September 2015. This river was hard to cross, and would have been out of the question without my wading staff.
See also advice on Outdoors SE for rivers.
In any case, whatever you do, DO NOT SET OFF WITHOUT A SATELLITE PHONE or at least a PLB. Renting a satellite phone for 2–3 weeks shouldn't cost more than in the order of ~£100, a reasonable addition to the budget of what will surely be a spectacular vacation. PLBs are cheaper, but may be harder to find.