I am tired of writing $\frac{\partial f}{\partial q} - \frac d {dt} \frac{\partial f}{\partial \dot q}$ many times, especially when my $f$ is actually a more complex expression. I have taken to writing this as $\Theta(f)$ in my own notes, but I am doing a project which will be presented to other people and I want to know if there's any precedent for notation for this. I have searched and found nothing: the closest I have found was the usual $\delta$-notation for the functional derivative of a functional, but I am searching for this specific operator.
So, is there any pre-existing notation for $f \mapsto \frac{\partial f}{\partial q} - \frac d {dt} \frac{\partial f}{\partial \dot q}$?
@EuYu's comment is sufficient and correct. Once you define $\delta$ as the functional differential, you can write $\delta F/\delta q$, or $F_q$ where it is unambiguous to do so. This essentially represents the Gâteaux derivative.
The notation $\delta f/\delta q$ in your comment does not make sense, as $f$ is a function so its domain cannot be a manifold.
If you do not want to introduce $F$ in your notation, I would go for what you have done, or rewrite the expression as $f_q-(f_{\dot q})'$ for simplicity (again where it is unambiguous). Otherwise there is no standard notation for the LHS of the Euler-Lagrange equation.