Twisted-arrow construction for 2-categories

While it is more explicit and combinatorial, this probably isn't precisely what you're looking for. It does seem relevant to your question, though. A year or so ago I worked out a 2-categorical analogue of the $(\infty,1)$ twisted arrow construction (I'm working with $(\infty,1)$-categories arising as Joyal fibrant replacements of the Duskin nerves of 2-categories). I initially tried the same approach Mike Schulman suggested in the comments, i.e., using Buckley's 2-categorical Grothendieck construction, but I found that a simpler construction was sufficient for my purposes. It's a quite straightforward generalization of the commutative (up to natural isomorphism) diagram $$ \require{AMScd} \begin{CD} \operatorname{Cat} @>{N}>> \operatorname{Set}_\Delta \\ @V{Tw}VV @VV{Tw}V \\ \operatorname{Cat} @>>{N}> \operatorname{Set}_\Delta \end{CD} $$ to 2-categories. The construction is a little ad-hoc, but the basic idea is the following: From a 2-category $\mathscr{C}$, construct a new 2-category $Tw_2(\mathscr{C})$ by letting

• Objects of $Tw_2(\mathscr{C})$ are morphisms of $\mathscr{C}$

• 1-morphisms in $Tw_2(\mathscr{C})$ from $f$ to $f^\prime$ consist of diagrams $$ \begin{array}{c c c } A & \overset{f}{\rightarrow} & B \\ h \downarrow\hspace{6pt} & \searrow & \hspace{6pt}\uparrow k \\ A^\prime & \underset{f^\prime}{\to} & B^\prime\\ \end{array} $$ and $$ \begin{array}{c c c } A & \overset{f}{\rightarrow} & B \\ h \downarrow\hspace{6pt} & \nearrow& \hspace{6pt}\uparrow k \\ A^\prime & \underset{f^\prime}{\to} & B^\prime\\ \end{array} $$ where each triangle commutes up to a (not nec. invertible) 2-morphism, and the 2-morphisms satisfy the obvious commutativity condition. (Note that, by definition, these are the 3-simplices of the Duskin nerve of $\mathscr{C}$.)

• 2-morphisms are given by (appropriately coherent) natural transformations of diagrams which are the identity on $f$ and $f^\prime$.

The resulting category $Tw_2(\mathscr{C})$ is only lax unital. However, with the appropriate nerve constructions, the diagram $$ \require{AMScd} \begin{CD} \operatorname{2Cat} @>{N}>> \operatorname{Set}_\Delta \\ @V{Tw_2}VV @VV{Tw}V \\ \operatorname{Lax2Cat} @>>{N}> \operatorname{Set}_\Delta \end{CD} $$ commutes up to natural isomorphism.

There is an obvious functor $$ Tw_2(\mathscr{C}) \to \mathscr{C}\times \mathscr{C}^{\operatorname{op}}, $$ though I haven't looked at its fibrancy properties yet.

Disclaimer: This is my first answer here, please let me know if I have ignored some protocol out of inexperience.