How many bins to include when calculating SNR from FFT?
The OP's question requires further details to provide a definitive answer, but the following will give the considerations involved. If the noise is white and stationary then the answer is clear in that we can simply use a power spectral density and work with the SNR as an SNR/Hz quantity. If the noise is not white (meaning the average power across all bins is not the same) then the concern will ultimately be with the measurement bandwidth of your final system that is providing the estimate of the quantity you are looking for. You would include all noise that is within this bandwidth as that is what will give you the statistical error in your result (which is why we are concerned with SNR in the first place). Usually the choice of bandwidth comes down to a time /frequency decision and the coherence time of your signal, and for how long it can be assumed to be stationary. For bandwidth limited signals that can be assumed to be stationary for a long time, then we can use a relatively longer averaging time (reducing the bandwidth down  to that of the signal, and therefore the amount of total noise in our result, as long as the noise is a spectral density and not a &quot;spur&quot; that exists at one frequency).
Ultimately to answer your question, determine accurately what the bandwidth is of your measurement. Use that bandwidth in the analysis where you predict the total noise and you will get a useful SNR. (This makes sense, consider the result you will get from your measurement system with a noise-free signal, and then consider the result you will get with noise alone-- the ratio will be your SNR that matters as it is based on the final result-- any noise that got filtered out by your measurement system does not apply).  This is analogous to a communication system where we will use the bandwidth of the signal, since the ideal receiver will have filtered out anything outside of that bandwidth but not reject any of the valuable signal. Your ideal measurement system should do the same, which means you will have interest in the spectral occupancy of your signal (and for how long it can be assumed to be stationary, which leads to concern with phase noise since that is typically not stationary... read on below).
You are likely doing an experiment where you are trying to estimate the presence of a specific signal, in the presence of noise, and ultimately will need to determine the best observation time in which to maximize your SNR (and therefore the accuracy of your estimate). Phase noise as your first measurement gave is a great indication of this as it is typically (if not always) NOT stationary over long durations, (and therefore for very long duration the typical power spectral density measurement is not valid). A much better measurement you can make that will allow you to optimize your estimation duration is the Allan Variance (also known as a two-sample Variance) since that will indicate the duration of time in which you can use an assumption of stationarity, and will then give you the averaging time that will maximize your SNR.
I go into more details of the Allan Variance and Allan Deviation for this particular application here: What determines the accuracy of the phase result in a DFT bin?