I seem to remember hearing this song on one of my daughter's music tapes (possibly from the "Wee Sing" folks.) When I try to replay it "in my mind's ear", I recall the "Goin' on a bear hunt", "I'm not afraid" parts, but I also remember the adventurers encountering various obstacles, and singing (in some order)

Can't go under it [can't go under it]
Can't go over it [can't go over it]
Can't go around it [can't go around it]
Guess we'll go through it!

An amalgam of the two versions you posted.