I'm afraid that I don't recognize this balance off hand. Would it be possible to see the balance removed? A lot of important details as to how it functions are obscured and I'm not sure I can make a recommendation without it out.
In the meantime, my best guess would be our Series 765. This has a similar looking hook on the bottom that yours does at the top, though that in itself is pretty weird. That hook really ought to be on the bottom. Part of me wonders if the original installers used this balance by mistake and just installed it upside to make it work. I really don't know, though. Without knowing more about the original and how it functions then I don't want to guess.
I know that you don't want to remove the balance at this point, but it may be necessary. There's not much else I can glean from these photos other than that it might be an upside-down version of our Series 765, which is less than an ideal recommendation.
If my theory above is accurate, and if someone did install the wrong balance the wrong way, then I wonder if something like our Series 770 might be a better fit for the window. This screws in at the top and attaches to a pivot shoe like yours with the terminal clip at the bottom, where the cord extends. That does seem to be how your current balance works, and the Series 770 would do a much better job of this than the Series 765.
However, there is the possibility that you have some other kind of balance I've never encountered before and it is working as intended, in which case using a different balance may just screw things up.
The pivot shoe, meanwhile, looks like an older version of our 15-046. You should compare your dimensions to ours to confirm, though, just to be safe.
