Echo removal from a chunk of PCM signal
Your $S_{new}=N-F$ requires that you know both $N$ and $F$.
But since $N=F*R$, with $*$ being convolution, and $R$ being the Room Impulse Response (RIR), you're solving the equivalent problem $S_{new} = F*R-F = F*(R-\delta)$, with $\delta$ being the delta dirac impulse or its discrete equivalent.
The hard part here is estimating $R$, which aligns nicely with your initial statement that you want to do echo detection in itself, not signal reconstruction. So, you're building a RIR estimator for itself, not for usage as part in an equalizer. That makes a lot of sense: Maybe a heuristic based on observing the MFCCs as they are, then &quot;cleaning them up&quot; in MFCC domain (simply nullifying things that shouldn't be there), then estimating a time-domain signal that would have these corrected MFCC signals, then using that as estimate $\hat F(t)$ for $F(t)$ and the received time-domain signal $N$ in a classical IR estimation algorithm, giving you a RIR estimate and thus an echo description.
Since convolution with the RIR is an LTI system, trying to do the estimation itself in MFCC domain feels like a bad idea – what is easy in time domain and linear frequency domain suddenly becomes hard: MFCC's are great if your system is not linear, but make working with linear systems harder!
If you insist on integrating MFCCs into your detection process, you could try to use them  as measure of speech quality, and feed that knowledge back into an adaptive time domain filter.
Generally, if I only had MFCCs, I'd try to reconstruct a time-domain signal from that, and then apply the mature theory of impulse response estimation on that – there's plenty of results that achieve optimum performance under some given conditions, and if you have one of these, whatever else you could do can only be worse.