If I recall correctly, we decided the character would be written out in January as having been found by her husband and she left Austin with him. It hasn't been mentioned explicitly in the game, I don't think, so we can be flexible on that. So it sounds like that could work perfectly for your character to have come here some time prior to January, been familiar with that character, the department, and the rest of the shelter leadership, and then taken over.
Pete lives and works at the UMCB, but he recently saw a patient associated with the Dog Park (though she wasn't living at the Dog Park at the time... longish story!) at the LBJ in December since that was the safest place for them to meet. Due to her being a member of that crew, she wouldn't have been allowed to enter the UMCB. Ever since then, he's been working with hardestofhearts (besties with jurisdoctor and another doctor at the UMCB) to secretly see patients who can't access the UMCB on the sly. Since the LBJ and the Dog Park are allies, that would probably also be something your character could do in the LBJ as well - either visit the Dog Park to provide care, or see Dog Park-allied folk at the LBJ itself. IDK! But I"m linking hardestofhearts to this comment too :)