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.