I've got a stream of digital IQ samples let's say at $F_s = 1 MSPS$ that consist of a narrowband signal, in awgn. Let's say the signal BW is ~ 100Hz. The narrowband signal's center (or base) frequency is variable in the range from -10 kHz to +10 kHz due to the doppler effect. For simplicity, it's fine to assume this is a simple linear sweep of the frequency offset.
I can imagine two simple ways of trying to visualize the frequency domain of the data. In method 1, I simply filter the data with a low-pass with cutoff of ~ 10 kHz and then perform an FFT and plot (or use welch/bartlett/etc). The FFT can be quite large (and give large processing gain) because I'm still operating at a high sample rate, and the latency of the computation won't be so bad).
In method 2, I could instead downsample (filter + decimate) to $F_s \sim 20 kSPS$. Again we perform an FFT and plot - however this time, because of the reduced sample rate we need to use a smaller sized FFT (mainly due to latency & not having enough samples available).
So my question is - does method 1 have any advantage here? Specifically - I'm interested in what would be most useful for resolving the narrowband signal when the SNR is poor & the frequency offset may be moving. In that case, it seems to me that there's a real problem with method #2 as the frequency may more easily sweep energy across multiple bins of the FFT. Taking that point further - would I be right in saying that the optimal option (ignoring computational complexity) would be to maximize the sample rate of the IQ stream, filter it to the BW I require, and perform large FFTs on that high sample rate without downsampling?