Hive beetles can get ahead of a hive if they are weakened, but one beetle on occasion may not be a concern. They enter and try to set up shop, but the bees can keep them harassed where they don't make headway. I had a hive last year in sun, but it sat on a deck which meant cool damp ground underneath. There were also a couple bushes next to the hive. It had beetles a lot later last year, but the were never allowed to do harm. This spring beetles still were in the hive, but I moved it to full sun on hard ground with no shade nearby, and the beetles went away. It's the only hive that had some beetles, and there were dozens.
About patience in a thread, I remember going through the same thing, and I am glad to be helpful, if I am actually helping. I hope I don't get all full of myself and all huffy about someone who doesn't get some of the basics yet.
I don't know what to think about the cell in the hive. Is it by chance an empty one that has the door shut back up? There's also a chance it went laying worker already and they are trying to make a queen with drone brood, but it's quite early for that. If a virgin queen mated and laid very early, they could have a cell, but that doesn't seam logical to happen.
Our flow ended last year a couple weeks after it started in early June. It wasn't too great because it didn't rain after mid May or so. I don't know exactly how long it would go, but I am assuming into July a bit if the weather and rain is good. All that sugar, you could feed if needed to keep them building up. We could get a good fall flow too, for wintering stores. Last option, you can combine them late in the summer if needed. No sense getting rid of a queen if you end up having them.
I am just trying to help with your hope in Queenlyness catching on. It sounds so noble.