He/she or what else could fit in a sentence referring to a transgender person?

Solution 1:

To be the most politically correct, use whichever pronoun the individual prefers. If the preference is unknown, one may assume that it corresponds to whichever gender they present; for example, if the person is biologically male but identifies as female, use she. It is impossible to tell from the context you have given us which is which in this case.

For what it's worth, referring to someone as "the transgender" is not really politically correct either; it would be better to rephrase so the transgender is understood to be only one aspect of the person's identity. The transgendered applicant would do, and then subsequent references could be made to the applicant.

There have been occasional efforts to introduce alternatives, or to use the singular they, but I would not say that there is any consensus on the matter.

Solution 2:

Apparently, the person in reference is currently so identified.

Transgender identities: While people self-identify as transgender, the transgender identity umbrella includes sometimes-overlapping categories. These include transsexual; …

…

Transgender Law and Employment Policy defined transgender as an expansive umbrella term including "transsexuals, transgenderists, cross dressers" and anyone transitioning. …
Transsexual: Transsexual people identify as a member of the sex opposite to that assigned at birth, and desire to live and be accepted as such. [emphasis added]