It is true that てめえ (てめぇ/てめー) is a contracted form of 手前, but they are used very differently today, so it's probably better to think of them as different words.
As you probably know, てめぇ is almost always a rough second-person pronoun, but it is occasionally used like a first-person pronoun meaning "myself" or "自分". This first-person usage is mainly found in fiction, but real people may use it in some set phrases. For example, てめえのケツはてめえで拭きます (lit. "I'll wipe my ass by myself") is an idiom meaning "I'll make up for my own mistake". Likewise てめぇの in Sanji's statement means "my own".
On the other hand, 手前【てまえ】 is fairly uncommon as a personal pronoun, but when it does refer to a person, it usually is polite and humble "I". Plural 手前共 (てまえども, "we") is occasionally used in modern Japanese, although it sounds old-fashioned and used only by a few businesspersons. This is a humble expression, and thus it never contracts to てめぇども. For example it's possible to say 手前共の店では着物を扱っております. 手前の尻は手前で拭います is a less dirty equivalent of てめえのケツは～. (It's still dirty, so use with care.)
