Is there any specifically masculine character for 他
I would suggest that there is no sense of gender in the Chinese languages since the very beginning; nor even today. We all know that the Chinese languages are very isolated and very analytic. The idea of gender, number, and case, in Romance languages may not be brought to the learning and using of the Chinese languages. Though I'm not perfectly confident, in the context of learning and using the Chinese languages, it might be a good practice to abandon the subconscious paradigm of fusing additional information into a word. If necessary, specify specifically.

As another user already mentioned, the invention of "她" was only a very recent activity (around 1900) by intelligentsia from English or Deutsch, and observable objections against it have arose recently (5-10 years) when the proposal of the abolition of gendered pronouns in the anglophone communities became popular.