lawson model 7700 impact window actual window size W 46.5 L 61"

I don't believe this is a Series 770 channel balance. Your balance has nylon fittings at the top and bottom, which the Series 770 lacks entirely. Furthermore, the Series 770 is only made for tilt windows, which yours does not appear to be.

In order to proceed, I will need a clear photo of the top and bottom nylon fittings. Would that be possible? Seeing the metal clip at the end of the cord would also be very helpful.

Great, thank you!  Based on what I see here, you have either our S380-30 or S385-30, depending on the depth of the bottom shoe. That's the only difference between them. I recommend measuring your shoe and referring to our dimensions to confirm the best match.

Comparison of the S378, S380 and S385 balance shoes.