I've discovered yet another method in this paper https://ejde.math.txstate.edu/conf-proc/21/k3/knowles.pdf (citing their earlier work, https://link.springer.com/article/10.1007/s002110050111). The derivative of the smoothed function, which they call $g$, is the same as the derivative of a modification $p = g + k$, where $k$ is a constant that raises the function to make it positive everywhere on the domain of interest. They then invoke Calculus of Variations to say the solution is at $−p''(x) + q_\text{opt}(x)p(x) = 0$, where $q_\text{opt}$ solves the optimization problem:
$$ \min_q \big[ E(q) = \int\limits_a^b (p'(x)−f'(x))^2 + q(x)(p(x)−f(x))^2dx \big]$$
This is similar to the optimization problems from TVR and Splines, but fancier. The authors take the Gateaux derivative of $E$ to step downhill. I'm honestly not sure anyone aside from Knowles uses this one, but worth some investigation.
