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, we obtain a notion of smooth, compactly supported functions on $G$. So the proof theorem Dixmier Malliavin extends to locally compact groups. See also my Phd thesis http://webdoc.sub.gwdg.de/diss/2012/palm/palm.pdf, where I discuss this in chapter 5, in particular Theorem 5.2.1. Approximation via Lie groups is explained in chapter 4.

As Yemon Choi points out, a necessity of the Plancherel theorem (check e.g. http://www.encyclopediaofmath.org/index.php/Unitary_representation) is that the $tr\; \pi (f\ast f^*)$ is well-defined for every $f \in L^1(G) \cap L^2(G)$ and a.e. $\pi$ with respect to the Plancherel measure $\mu$. 
*So, I suggest to change your definition of trace class function to mean this*. 
The Plancherel theorem states then for $G$ be a unimodular separable locally compact group of type I
$$ \int\limits_G  |f(g)|^2 d g = \int\limits_{unitary \; dual} tr\; \pi( f \ast f^*) d \mu(\pi).$$
There exist extension, which drop unimodular, type I or seperable, but I am no expert on this.

You can furthermore put a (Fell-)topology on the unitary dual and so on, but I don't know wheter $\pi \mapsto \pi(\phi)$ will then be a continuous function!? Even a smooth structure via the approximation-by-lie-groups-business. 


Using Dixmier-Malliavin plus the polarization identity you obtain for $f \in C_c^\infty(G)$  (Theorem 6.2.6, pg.83 in my thesis) the following variant of the Plancherel theorem
$$ f(1) = \int\limits_{unitary \; dual} tr\; \pi( f) d \mu(\pi).$$

So the answer is affirmative for $G$ a unimodular separable locally compact group of type I or any locally compact group with a suitable Plancherel theorem. For non-type-I groups, the Plancherel measure will not be unique, so I am sceptical about a direct generalization omitting that notion. I think that neither being unimodular nor seperable is a crucial hypothesis.


For reductive groups over local fields, you can replace a.e. w.r.t. to Plancherel measure by every smooth, admissible representation, but I guess you are well aware of that. The notion of smooth in the sense I indicate above coincides with the usual one in the non-archimedean world/totally disconnected groups. Be careful, general Lie groups are not type I. For the adelic points of a reductive group over a global field, you need an argument along the lines of http://mathoverflow.net/questions/116243/is-a-reductive-adelic-group-a-type-i-group, but the situation is similar to that of reductive local groups.