What translation did you get this wording from?

It's not found in KJV (old school Protestants ), Douay Rheims, NIV(mega churches, some Baptists, many others), or NASB (mainly used by Catholics) translations.

It's not a translation. It's an allusion. Commonly used by evangelicals. You're over thinking it.

Source please?

I grew up in an evangelical home and went to church every Sunday - and have never heard of "may his tribe increase" aside from that poem.

It's not just you. Googling "may his tribe increase" and "Genesis 22:17" only finds this thread.