Bounding the difference of random variables by coupling. Suppose we have two probability densities differing by atmost $\delta$. Is it possible to use coupling to have two random variables with the above two densities differing by less than $\delta$? I also have that the densities are differentiable k times with all the derivatives being Lipschitz continuous. And the densities have compact support

Certainly not without further conditions. For example, maybe the random variables are spread out over such large intervals that their densities are always less than $\delta/2$.

EDIT: Consider densities $$\eqalign{f(x) &= \cases{1/2 & for $0 \le x < 1/2$ or $1 \le x < 3/2$\cr 0 & otherwise\cr}\cr g(x) &= \cases{1/2 + \delta & for $0 \le x < 1/2$\cr 1/2 - \delta & for $1 \le x < 3/2$\cr 0 & otherwise\cr}}$$

If $X$ and $Y$ are random variables with these densities, $P(X \ge 1) = 1/2$ while $P(Y \le 1/2) > 1/2$, so no matter how small $\delta$ is, $P(X - Y \ge 1/2) > 0$.

These densities are discontinuous at $0, 1/2, 1, 3/2$, but it's easy to construct similar examples that are smooth.