You have the right to use whatever word you like. You have the right to use offensive words. You have hundreds of thousands of words to chose from.

"Anorexic" is an odd choice. Dealing with this purely from a linguistic point of view, and ignoring the "you might offend a few people" point of view, there are some unfortunate connotations.

Anorexia is a serious mental health problem. "The mentally ill start-up"?

Anorexia is, when we adjust for age etc, the most lethal mental health problem. About 1 in 5 anorexics die from it. "The at risk of death start-up"?

Anorexia in its severe forms is very distracting for the person. It takes a lot of effort to avoid eating; it takes a lot of concentration to count calories; it takes a lot of effort to 'body check' before you leave the house. "The distracted start-up"?

Anorexia leaves the person weak, susceptible to illness, and unable to do much. "The frail start-up"?

There are better words for "ultra lean".