Can he say “Ana faatir (I am not fasting)” if he is not fasting?

If a person tells someone that he is not fasting, he should say Ana muftir, because muftir is the active particle of the verb aftara (to break the fast); the verbal known is al-fitr, which means breaking the fast. 

Al-Jawhari said: Aftara al-saa’im (the fasting person broke his fast); the verbal noun is fitr. [And he gave further examples of how the words fitr and muftir are used].

End quote. Al-Sihaah fi’l-Lughah, 2/47 

The word faatir is the active participle of the verb fatara, and the verbal noun is al-fatr, which means initiating or creating. 

Al-Zubaydi said: fatara al-shay’ means he began the thing. Ibn ‘Abbaas said: I did not understand what faatir al-samawaati wa’l-ard (Creator of the heavens and the earth) meant until two Bedouins came to me with a dispute about a well, and one of them said Ana fatartuha (I started it), meaning that he was the one who dug it first. 

Abu’l-‘Abbaas said that he heard Ibn al-A‘raabi say: Ana awwal man fatara hadha (I am the first one who started this), meaning the first one to begin it. End quote. Taaj al-‘Aroos, 1/3347 

This indicates that the verb fatara, and its active participle faatir, are not among the words that are used exclusively in reference to Allah or in conjunction with He name, may He be exalted. Rather these words may be used to refer to others, even though they are rarely used in the manner even with regard to one who starts something new; such a person is rarely described as faatir. 

The mistake here is merely a linguistic error.  If the dialect of the speaker usually uses the word faatir to refer to one who is not fasting, it does not seem that there is anything wrong with using it in this manner. 

And Allah knows best.