Suppose that we consider the function $F: \mathbb{R}^d \to \mathbb{R}$, defined by $$ F(x)= \begin{cases} \widetilde{F} (x), & x \in K, \\ 0, & x \not\in K. \end{cases} $$
where $K$ is a compact subset of $\mathbb{R}^d$ and that $\widetilde{F}: K \to \mathbb{R}$ is in $C^{\infty}$. To mollify $F$, a common technique is to convolve with a sequence of mollifers $\{ \varphi_{\varepsilon} \}_{\varepsilon>0}$, such as those corresponding to the bump function. $$F_{\varepsilon} := \varphi_{\varepsilon} \ast F$$ does the job.
However, with most mollifiers, $\| F_{\varepsilon} \|_{\text{Lip}}$ goes to infinity as $\varepsilon \to 0^{+}$. Is there a method of smoothing that can prevent this from happening?
It's impossible to approximate pointwise a discontinuous function by a sequence of Lipschitz functions with uniformly bounded Lipschitz constant. You can prove easily that if $f_n \to f$ pointwise and the Lipschitz constants of $f_n$ are all bounded by $L$, then $f$ is Lipshitz with constant $L$. Just note that
$$|f_n(x) - f_n(y)| \leq L|x-y|$$
for all $x,y,n$, and send $n\to \infty$ to find that
$$|f(x) - f(y)|\leq L|x-y|.$$