Dalvik.
I admit to being a sleazy heli access type... But look into
Bergmenn for guiding. At least to get your feet wet. They know the place inside and out. Local knowledge (and facilities) is beneficial not just because of optimized access to snow & terrain, but other cool stuff in the area. Bonus - a couple folks there have deep WA state connections.
Most years, early May will offer lots of good snow. This year would not have. A couple years ago you could step out the door at Karlsa Lodge, walk across the street and be on snow - in early June. So early may is a good bet. But as always, you get what you get.
Iceland has gotten fairly pricey in recent years. Rekjavik doubly so. Places like Dalvik are a bastion of relative sanity. I'd guess (and it is only a guess) that you could stay a night in Dalvik (or maybe Akureyri), and join a good guided group, and knock back a few Kaldis and coffees for what just well located basic lodging in Reykjavik would cost for a night.
You won't go wrong with the food at Akureyri Backpacker (including one of the better breakfasts/brunch in the country). Good food can be had at either the Dalvik or Akureyri Culture Houses (I think Dalvik might be lunch only). The Dalvik coffee shop is great (think afternoon waffles). Ask about the tales of the 3 brothers pictured on the wall. Might remind you of polyass

Last I knew, it was also a good place to rub shoulders with other touring groups. Rub 23 is a good higher end eatery in Akureyri ("sushi pizza" all the way). Not much in the way of dinner places in Dalvik.
Drive or fly is up to you. Should be pretty easy driving that time of year. Allow 6 hours or so from Keflavik. There is a hot dog stop at the big N1 about half way (IIRC). OTOH, you can snag a bus from Keflavik to the Reykjavik domestic airport. The flight is an easy 45 minute hop - so land at 6:30 or so in KEF and you can be in Akureyri by ten or so. The domestic airport is more like a bus stop than an airport - check in 15 minutes before your flight - and it is five to ten minutes from downtown in bad traffic. Cars can also be picked up in Akureyri. Car rentals are expensive. So another thing to think about (again)- for the cost of basic lodging and a semi-decent AWD (and you want AWD), and meals, you might just about get a package from Bergmenn. Not totally sure and not saying that is the only way to go - but it is worth looking at all the the permutations of all the costs. Though admittedly I just plain like like & respect those guys.... I usually get to Iceland once a year & my head just spins at the rental car counter these days. If you rent, check the car out carefully before heading out - the place is hard on rentals and you want to be sure that nothing is falling off....
The North is great. It is hard to imagine not having a super time there. The other thing to note is that flexibility is both good and hard. It is good because things are weather dependent. It'd suck to sit out a bad day or two and then be locked into heading to Reykjavik on your first potentially great day. OTOH, it is now hard to do last minute bookings or cancellations - especially in Reykjavik. I have no magic words of advice - other than staying as flexible as possible is good.
And yeah - I should probably just have said "search function"... Clearly too much time on my hands today.
Oh - some of the sights in the south are worthy. If you are going to spend a few days in Reykjavik - you could do worse than book a tourist tour to the waterfalls and Glacier Lagoon. Or make a long day's drive of it (days are getting long by then).
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