If this "sentence" seemed difficult to understand, it would be mostly because it was a spoken sentence and not a written one. At least, the vocab level is fairly intermediate.
I have stated this here a few times before without much success but will state it again. Try to spot the "hidden" commas (and even periods at times).
君社長ともなれば = "Even if you become president"?
Nope, that is not what it means. It is not 君 who becomes or will become the president. The speaker is the president. Let us use a comma to illlustrate the grammar used at the beginning.
「何しろ君社長ともなれば」 ⇒ 「何しろ、君、社長ともなれば」
"At any rate, you see, when one becomes a (company) president, ~~"
「君」 here is only addressing the listener. For the advanced learner, 「君」 in this usage is often, if not always, pronounced differently than 「君」 as a regular pronoun. It is often pronounced 「きみぃ」.
The middle part of your TL attempt actually looks good, so let us move on right to the last part.
「つきあいきれんよ」, of course, means the same thing as 「付き合いきれないよ」
"Verb in Continuative Form + きる" means "to do something thoroughly or completely".
「つきあいきれんよ」 would therefore mean "I cannot keep up with (them)."
Back to adding the hidden commas and periods... The following would be one (safe) way to do it.
「何しろ、君、社長ともなれば、年始の挨拶に来る客が入れ代わり立ち代わり。(or a comma) とてもじゃないがつきあいきれんよ。」
My own TL:
"At any rate, you see, when one becomes a company president, guests just swarm in one after the other for the New Year's greetings. I simply couldn't keep up with them."