If you have a reduced model with $H_0:\beta_1 = 1$, $H_a: \beta_1 \neq 1$, then the reduced model is: $$Y_i = 1X_i + \beta_0 + \epsilon_i$$
Are the degrees of freedom for the error term SSE $n-1$?
You would have $n-1$ degrees of freedom for the error term in the null model.
I think a caveat may be needed if you want to do an F-test. With the usual assumptions about normality, independence, and homoskedasticity, if you were testing the null hypothesis that $\beta_1=0$, then the F-statistic would have an F-distribution if the null hypothesis is true. However, that conclusion relies not only on the fact that the numerator and denominator would each have a chi-square distribution, but also that they are independent. If I can believe some tentative stuff I just did, if the null hypothesis is $\beta_1=1$, then the numerator and denominator in the F-statistic would not be independent. I'm going to need to look at this more closely to see what should be done about it. It should be possible to derive a likelihood-ratio test. With the more usual null hypothesis, that would be the usual F-test. With this null hypothesis, it would be different.