While color and spectral type are definitely heavily related by temperatures, the way that spectral types originally came to be is (by the name) according to spectra, and so without the spectra it could be difficult to classify the star completely (e.g. into something like G5 with its luminosity class) but it seems like you’re looking more for a general idea, which could be obtained.
One way that isn’t necessarily a flux approach but could be used if you just really want spectral types is by going to MAST and seeing if there are other observations done (like by a spectrograph) that could more precisely narrow down the spectral type.
Another option if either you can’t find anything on MAST or are curious about the flux approach is to just find a table of some well defined spectral types and their corresponding B-V’s or some other related color mechanism (I imagine the type of color indicator might vary depending on what kind of stars you’re looking at). I’ve seen these tables before, most notably in the back of a textbook I have I believe, and so if you can’t easily find them somewhere let me know and I can find an exact source for the one I’ve seen. This is obviously filter dependent, so care would need to be taken to make sure you’re comparing measurements from the same types of filters.
This last method isn’t very precise, but neither is determining spectral type by color alone; it’s a good indicator for general classification, but harder for more specific classifications that require spectra.