In fact the pronunciation is phonemically /bɛʁg.sɔn/.
The final is not nasalized because it's considered a foreign name. Foreign names are generally read more or less letter by letter, although there is considerable variation depending on the name and on the speaker. Until the 18th century (I think), foreign names used to be transcribed into French, then pronounced using the normal rules of French. These days, for names that were originally in Latin script, the spelling is not adapted but the name is pronounced differently. For example Lisbonne, Francfort, Rome, Milan, Bombay, Pékin are pronounced the way they're spelled, which is different from the original name. Names such as Dublin and Berlin are spelled like the native name, but pronounced as if they were French names, with a nasal vowel [-ɛ̃]. More recent imports are pronounced in a way that approximates the original pronunciation slightly more closely: Washington [wa.ʃiŋ.tɔn] or [wa.ʃin.tɔn], Medellín [me.de.lin] or [me.dɛ.lin], Amman [a.man]...
With people's names, an added factor is that it is to some extent a choice of the people themselves. It can happen that different members of the family pronounce the name differently. For all I know, Henri Bergson might have had French cousins who pronounced their name [bɛʁgsɔ̃] or who spelled it Bergsonne.
As for the
g, I don't know how Bergson himself pronounced it, but both my philosophy teacher and the 1936 interviewer found by petitrien pronounce something close to /bɛʁg.sɔn/. The /g/ sound in French is nominally a voiced velar stop, while /k/ is a voiceless velar stop. However, the transition between the voiced /g/ and the voiceless /s/ is difficult, so its phonetic realization is somewhat devoiced. (This is the reason why
x is pronounced either /ks/ or /gz/: either both consonants are voiced or both are voiceless.) The /g/ in Bergson is still perceived as a /g/, though, not as a /k/, even though it is phonetically somewhere in the middle.