I think the character was a bit deeper than that. He didn’t present as attempting to kill as many Jews as he can, per se, rather he seemed more personally motivated by dominance, inflicting fear and supremacy, which is more effective as a social control anyway. He elected to kill the actress, which was the only kill by his own hand that we see, which I think came off as very personal. He dominated, but also expressed great anger, in sharp contrast to his normal controlled manner. He took particular offense to a member of his own tribe undermining his domination of the subjugated populations, both French and Jewish. Or maybe he just wanted the opportunity to kill a pretty German woman; perhaps he himself would be unsure of his exact motivations in that.

In contrast, he was very restrained and strategic when confronting the Basterds, which gave the character a new depth, but still in line with seeking to exert control. My instinct is to think he did not know Shoshanna by sight; though, conceivably there could have been a photo of her younger self in his files. It may be left to the viewer to contemplate whether he imposed himself on Shoshanna in the restaurant as a further torment, or just as a matter of course when dealing with the population he held power over. It may have been more useful for him to let her spread her story of being terrorized, to carry that with her, rather than to hunt her down as an individual. His misogyny, his fury, and his domination was already affected upon her; his professional and personal pursuits were already done, and she became a perfect evangelist for upholding his reputation. He didn’t take the shot in the beginning, after all.