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You can come down safely in the camp of wheelbarrow as the correct spelling. The etymology has nothing to do with barrel, despite the common misunderstanding which results in the pronunciation and spelling of wheelbarrel.

The Online Etymological Dictionary gives this for the barrow part of the word, which is a word unto itself:

barrow (1)
"vehicle for carrying a load," c.1300, barewe, probably from an unrecorded O.E. *bearwe "basket, barrow," from beran "to bear, to carry" (see bear (v.)).

Barrow

Webster gives this definition for the rare word barrow (in this sense — there are other independent entries for the word barrow):

Main Entry: (3) barrow
Function: noun
Etymology: Middle English barew, from Old English bearwe; akin to Old English beran to carry — more at bear
Date: before 12th century
1 a : handbarrow b : wheelbarrow
2 : a cart with a shallow box body, two wheels, and shafts for pushing it

Barrow