...obviously this designer was never read to as a child, jumped rope, or had his mother/father/guardian sing to him/her/questionable.

well "a bear" could be a refrence from that he was not literally a bear, but rather bare. this rhyme was made in the 1800's, and it was actually referring to a capybera, so "a bear" could mean that Fuzzy Wuzzy was bald, but since it was probably not written down for decades, the "Fuzzy Wuzzy was all a bare" must have gotten written down as "Fuzzy Wuzzy WAS a bear."
Another assumption is that "a bear" is in refrence to a capybera. They were sometimes nicknamed "Bera." To an outsider who did not know this, they may have left off the "a" on "bera" and mistook it for "bear."