Don't rush it. I posted a while ago that the eggs should soak a minimum of 2 weeks. I tried one at exactly 2 weeks and it was OK. I then waited another week and they were much, much better. At a month, they are awesome.
I think you can safely pull them out of the frig at 2 weeks and store at room temp. That was the whole point of pickling eggs back in the day. Shelf stable at room temp. I can't begin to estimate the number of pickled eggs that I ate at Chet's Friendly Bar in Point when I was in college and those were just on the shelf behind the bar. And I'm still here! The pickling would probably go faster at room temp. I just have room in the beverage frig in the garage so that's where mine are.
I'd try that recipe with the pickled beets. I did a few quarts of beets last fall.
I fired up a batch of eggs with the recipe I posted a while back. My wife laughs when she sees me taking off the cover and smelling them. Wednesday marks two weeks and then it's on!
I grew up in South Central PA and the only pickled eggs were called red beet eggs. We use vinegar, sugar, a few pickling spices, sliced onion, and sliced pickled beets. The eggs sit in the beet juice and are perfect at about 5 days. You want to have the egg whites red, and the yolk still yellow. Yum!!
If anyone wants the full recipe I would be happy to oblige in a message.