Is there any specifically masculine character for 他 The pronoun 他 (ta1) for he/she/it has been used since at least the Ming dynasty and has had a number of variants, such as 牠 for animals and 祂 for deities. In the 20th century the feminine version 她 was introduced to imitate western languages, and is now used to refer to female persons in almost all Chinese publications.

Since 他 was originally gender-neutral and uses the generic "human" radical 人, while 她 is explicitly gendered, this seems somewhat asymmetrical. I'd like to know if there has ever been a character proposed that would represent the pronoun "ta1" only when talking about male persons.

I know that the Chinese language has other 3rd person pronouns, some of which are gendered, but I'm interested in ta1 specifically.
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.