Is the expression "bête chevronné" correct for "seasoned/experienced beast"? I was told I need an extra e.
You do. Adjectives agree with the gender (and number) of the nouns they qualify. Since bête is a feminine noun, the feminine form of the adjective is required, which in this case is spelled chevronnée, with an extra e.
The agreement is purely grammatical as “bête” (feminine) may figuratively refer to a person or live being, or entity of any biological or attributed gender.
"Chevronné(e)" is used for persons, especially in the context of their professionnalism, and at that, only when their profession involves a rather important amount of knowledge; one will not ever say something like "une caissière chevronnée"; it means "fully skilled", "well versed in the subtleties of one's profession"; you could say , however, "un charpentier chevronné". It seems that what you mean is rather "une bête endurcie", that is a strong beast, trained to bear a burden without failing; in any case this corresponds more to "seasoned" than "experienced".
The added e is justified by the gender of "bête", which is a feminine word in French; in French there is a rule of agreement of a noun with its adjective, in gender and in number; e is added to obtain the feminine form of the adjective and s the plural; thus, if the beast were a donkey you'd write "un âne endurci ("âne" is masculine, so, no e); if there were several beasts you'd have to say "plusieurs bêtes endurcies" (first e and then s, never the other way).