I have a dataset that shows how to arrive at a measurement two
different ways. (an old clunky way of doing it and my new, faster way)
There is very good correlation between the two methods ~.91 r-squared.
There is a regression that was published a while ago in a similar
study that is different- but matches my data very well. How do I show
that the regression equation found by this similar study is adequate
for use with my data as well?

This other regression is empirically robust and accepted in the
community so I would like to show that the relationship holds up to my
new method. There are also some other regressions that- when
superimposed on my data- do not match up as well and I would like to
have a metric to show how each of them compares to the least squares
regression and the empirically robust regression.

A good and easy way to see the correlation between your method and the
accepted method is to apply them both to a totally different set of
data, one that would intentionally not fit very well at all, and see
what kind of error values you get with both methods. If they get very
similar results (despite being outrageous), then both methods are a
good correlation and you're safe using your own.

I was hoping for something more quantitative. Akin to r-square value.
I found an exel sheet that calculates the sum of squared errors. The
regression I would like to compare has a low SSE, almost as low as the
true least squares regression line- but by itself isn't recognized as
a measure of correlation.

Perhaps I am asking to show how a model fits a dataset. I am not
versed in statistics enough to know if my vocabulary/semantics are
correct.