First, FDR basically had a mandate.  Congress was overwhelmingly Democrat, so that dog won't hunt.

Second, he wouldn't have pushed to pack the court of the SCOTUS hadn't ruled against some of his New Deal policies.  Hence the wish to pack the court. 

FDR proposed the court packing in 1937:

 Democrats had basically a stranglehold on both houses of Congress, so if there was any obstruction to his New Deal programs, it would have had to be bipartisan.

The 75th Congress: Democrats appeared to have a 69-23-2-1-1 in the Senate(keep in mind this was before Alaska and Hawaii were states) , and 334 seats (with only 88 Republicans) in the House.  That was easily enough to overcome any Republican "obstructionism" in those days.

Blaming Republican obstructionism for every Democrat administration may sound good, but it doesn't hold up to reality, especially in FDR's administration when his party had a mandate.