Another of those pro-'Rat, pro-Roosevelt myths that seem to get passed down from generation to generation in too many Jewish families!
Fact is that Democrats had such wide majorities in Congress following the Roosevelt landslide in 1932 and continuing through the remainder of the 1930s (although Republicans did make significant gains in the 1938 mid term elections) that Republicans couldn't block anything, even if they tried. The so-called "isolationist" movement, which was associated with restricting immigration, was bipartisan.
I was careful to add that I was repeating what I was told, and hadn’t verified it. You may be quite right.