Iopened my hives today for an inspection. I have 4 and 3 have very little if any stores. I only took honey from 1 this year as it was a wet summer.
I have 1 hive that was a swarm I captured last year. They overwintered fine and I requeened them in April. I know I physically destroyed the old queen as I had witnesses. After installing the new queen I relocated them and all was fine. The new queen was laying like crazy but they kept building superscedure cells. I destroyed what I thought was all of them. Even with no flow on I kept noticing a lot of brood in this hive, it is 2 brood boxes deep. Today while inspecting I pulled a frame on the top box and found my store bought marked queen. She's big and healthy looking. I pulled the top box off to ck. the bottom and while checking for any honey stores I couldn't believe my eyes. There in the bottom box was a queen as big as the other, this one is not marked and looks healthy too. This hive has plenty of brood in both boxes but not much stored food. I am short on woodware right now but could probably come up with what I need to split if this is adviseable. My biggest problem with splitting or leaving them alone is they have very little honey. They had built up some but have obviously used it feeding.
Put the feed bag on NOW and keep it going.
You need a min. of 120 lbs. of hive and stores to make it through the winter. Maybe less in your area, maybe more for a two queen colony. Depends.
You have a natural two queen colony. Not unusual at all. They built a supercedure cell, raised a new queen, but haven't gotten fed up enough with the old one to get rid of her. My guess is there will only be one next spring.
I would try to feed so they can make it through the winter.
If I'm going to have to feed them would I be better off splitting them?
You've got a hive that's superseded but hasn't got round to dumping the old queen yet. Happens all the time with some strains. Splitting could be risky; they may well raise a new replacement queen, but something could go wrong, and you don't know how much longer the old one will last. If it was me, I'd feed the hive and let well alone.
I agree with Robert. They will get rid of her eventually and splitting will just make two weak hives going into winter one of which has a failing queen.
I would leave the bees alone to sort it out.