Channel balance

The balances will be the same between left and right. Choose Series 380 or Series 385, depending on which shoe you have.

For the 1-3/32" shoe shown below, choose S385-18. For the 1" shoe, choose S380-18. I'm leaning toward the smaller shoe, but you can't always eyeball it. Measure up to be sure.

Both of these balances are available as "1740."

Comparison of 15-004 and 15-005

My initial advice is that the 1" shoe can sometimes wear down to below 1" in depth, and that the shape of your shoe is not different enough to make a difference.

To confirm that this will work, what is the exact depth, if not 1"?

That should be fine. 1/8" is unlikely to make a difference. If you really end up needing to, you could file the plastic down until the shoe fits, but I don't believe that will be needed. There's no 7/8" shoe on the market, so this is your best bet.