Here is an alternative answer (also based on the control of the growth at infinity): simply use Lebesgue's Dominated Convergence theorem:
Note first that the convergence  $\|f_n-f\|\to 0$ in your $Lip_0(\mathbb R^d)$ space immediately implies pointwise a.e. convergence,
$$
f_n(x)u(x)\to f(x)u(x)
\qquad a.e.
$$
In order to apply the DCT we need a uniform $L^1$ upper bound.
For this note that the Lipschitz norm controls the growth at infinity, hence $
|f_n(x)|\leq \|f_n\|\,|x|\leq 2 \|f\|\, |x|
\qquad \forall x$ uniformaly in $n$.
In particular given your assumptions on $u$ we get
$$
|f_n(x)\, u(x)|\leq 2 \|f\|\, |x|\, u(x)\in L^1
$$
and the result follows.