You state Duflo Labesse in a slightly wrong manner. You need a finite sum of convolution products. I thought it was due to Dixmier Malliavin, though. 

Writing general G as an projective limit of Lie groups, it extends in a suitable manner to the smooth, compactly supported functions on G. So your proof extends to type 1 groups. See also my Phd thesis, where I discuss this as well.

My intuition about non-type 1 groups is zero:-(