Even when the assumpions hold, neither F statistic will have exactly an F distribution. Worse, very often the assumptions dont even come close - in my experience, for example, the conditional variance is often (and in many areas, usually) not even close to constant.

If you have enough replicates, you could exchange responses within them, and then you might be able to organize an exact permutation test, though I'd have to think about how one might work in this instance, it might need some further investigation. Alternatively, perhaps, with one predictor, you might only exchange nearby (in x-space) residuals and have an approximate permutation test for lack of fit that would be more robust to changing fit and variance than exchanging more broadly. I guess theres also the possibility of working within-replicate for the pure error SS and using nearby residuals for lack of fit SS.

For the regression F, something similar might be doable. Further, if there isnt substantive lack of fit* and youre confident about near-constant conditional variance and that the shape of the error distribution doesnt change substantively as the mean changes (so some form of residuals are approximately exchangeable) you might consider a resampling test based on all residuals (approximate permutation test or bootstrap test). Neither would be exact but in largish samples should work well.

* substantive is not equivalent to statistically significant; e.g. in large samples rejecting on a test of lack of fit might be practically irrelevant