Ba`al
It is true that ba`al1 usually means something from owner/lord/husband categories. However, there are more possibilities. Gesenius' 18th ed. lists also "inhabitant" (and, of course, the deity Ba`al). However, in poetic language, meaning I.4 is often appropriate:
häufig i. Zssetzg. m. einem Subst. z. Bez. einer Eigenschaft od. d. Teilhabe an etw.
meaning
häufig in Zusammensetzung mit einem Substantif zum Bezeichnung einer Eigenschaft oder die Teilnahme an etwas
meaning
frequently in combination with a substantive to mean a property or the participation to something
Prov. 22:24 is given as an example, בעל אף "zornmütig" "irascible". For Prov. 23:2, Gesenius has "gierig, heißhungrig" "greedy, ravenous".
Personal name
Another thought was, since the Hebrew doesn't have all the verboseness we Americans seem to love, and instead just has the phrase "lord appetite" (as it could be translated using Strong's), could this be a descriptive phrase?
While it is of course difficult to prove that this was not possible, no examples of such usage (also with other nouns) come to mind. When a king/queen is addressed, in the Hebrew for "king/queen X" the word "king/queen" usually has a definite article ה "the" (e.g., in the king's speech in Esther 7:2). This makes sense, because personal names are determined and the nouns must agree in definiteness. In this proverb there is no article which would be a case against this hypothetical translation.
The Hebrew is not problematic in this way, it is a very common example of a construct chain, a chain of nouns in the genitive relation (e.g. "the book of the king of Israel") for which Hebrew does not use a word for "of" but signals by word endings that words are part of a construct chain. However, in the masculine singular (like ba`al) this case is not distinguishable from the normal state of the noun, so we can theoretically argue either way — though, again, the text is not problematic.
Meaning
The exact meaning of the word נפשׁ is strongly debated, still, but it means something in the direction of "life force", "breath", etc. In this case I would understand it as having the will to act quickly without considering first. Thus the meaning of the first three verses of the chapter is more or less: (1) don't take what rich people offer you for granted, "consider it diligently"; (2) even more so if you are quick to act; (3) do not trust their "dainties".
Wycliffe's main contribution is to put a sentence break in the middle of v. 2, thus getting two tricola rather than three bicola, in the sense of "If you master your appetite (נפשׁ), then do not trust their dainties". This allows reading ba`al in the sense of "owner", "master". There is something to be said for this. In particular, the verbal form of "put a knife to thy throat" is a wəqaṭal which continues a previous imperative ("consider", v. 1). However, this is not unthinkable in a new bicolon, and it does require breaking up verse 2. It is not required to resolve the meaning of ba`al, since the meaning discussed above is widely attested.
1: When transliterating, usually the `ayin is indicated with a backtick (`) and the 'aleph with an apostrophe ('). In this case we have an `ayin, so it would be ba`al (although in the current SE font the difference is hard to see).
