If the carburetor works well in first and second (also the timing) then it is also good in third.
I would check:
(A) fuel filter
(B) possible clogged tank vent
(C) leak-down test on fuel lines from tank to pump
(D) fuel pump volume test
(E) (stretching here) possible worn fuel pump lobe on camshaft
(F) minor kink in fuel line restricting volume
Jon.