Since you started with the quasi-categorical version of the twisted arrow category (and not the complete Segal space version), maybe $\Theta_2$ isn't quite the next step. Instead, let's see what happens if we keep things as close to simplicial sets as we can- say, the complicial route a la Verity. For $n=2$ it's possible to make due with less, and we have Lurie's model of $(\infty,2)$-categories via special types of scaled simplicial sets, i.e. a simplicial set $S$ together with a collection $T \subset S_2$ of *thin* 2-simplices (which we think of as being 'the invertible 2-morphisms') that contain all the degenerate 2-simplices.

If $\mathcal{C}$ is an $(\infty,2)$-category, we expect the `hom'-functor to be some type of thing like $\mathcal{C}^{op} \times \mathcal{C} \to \mathsf{Cat}_{\infty}$ where the target is the $(\infty,2)$-category of $\infty$-categories. I claim that the appropriate candidate for the twisted arrow category is... **the twisted arrow category (again)**!

Let me say what I mean more precisely. Given a scaled simplicial set $(S, T)$ form the *marked* simplicial set $(\mathsf{TwArr}(S), M)$ where $\mathsf{TwArr}(S)_n = S([n]^{op} \star [n])$ and a 1-simplex is marked if and only if the two 2-simplices making up the face of the square are thin. This thing has a projection map to $S^{op} \times S$. I claim that when $S$ is fibrant (i.e. an $(\infty,2)$-category) this is an appropriate sort of fibration which is classified by the lax functor $\mathrm{Map}(-,-): S^{op} \times S \to \mathsf{Cat}_{\infty}$ mentioned above. I'll try to sketch how that goes.

But first, as a sanity check, notice that the fiber above vertices $(X,Y)$ has 0-simplices given by 1-morphisms $X \to Y$ and $1$-simplices given by a little square. Collapsing the edges $(X=X)$ and $(Y=Y)$ leaves you with a little disk with a line through the middle- all arrows oriented left to right. Keeping track of orientations of the 2-simplices tells you that the two 2-arrows glue together to gives a 2-arrow from the bottom to the top (or top to bottom depending on how you drew this.) This 2-morphism need not be thin, so we are seeing the 'category of 2-morphisms'. Not rigorous, but a little reassuring.

- Lurie proves a version of straightening/unstraightening which produces, for a scaled simplicial set $X$, a Quillen equivalence between a certain model structure on $\left(\mathrm{Set}^{+}_{\Delta}\right)_{/X}$ and a certain model structure on $\mathrm{Fun}(\mathscr{C}^{sc}(X), \mathrm{Set}_{\Delta}^{+})$ (where $\mathscr{C}^{sc}$ is a scaled version of $\mathfrak{C}$). Passing to a scaled version of the nerve, if $X$ is fibrant, produces a map of $(\infty,2)$-categories $X \to \mathsf{Cat}_{\infty}$.
- The first condition for fibrancy here is that we have an inner fibration on the underlying simplicial sets. I'll skip saying anything about this... you could draw some pictures for $\Lambda^2_1 \subset \Delta^2$ and convince yourself that if the definition of $(\infty,2)$-category says you can 'compose' 2-morphisms in an essentially unique way then you'll be ok. And it does!
- The next condition for fibrancy is that the marked edges coincide with the locally $p$-cocartesian edges. Intuitively, suppose we have a morphism $(X,Y) \to (X', Y')$ and we've chosen a lift $X \to Y$ of the object $(X,Y)$. Then a morphism covering the given one is the data of a 1-morphism and a 2-morphism from that to the composite $X' \to X \to Y \to Y'$. The universal choice of such is to take "the actual" composite and have the 2-morphism be the "identity". Of course, we're in a weak world, so to do this for real I'd have to tell you more about the definition of an $(\infty,2)$-category.
- Finally, we need to know that $p$ restricts to a cocartesian fibration along any thin 2-simplex. To check it, you need to know that composing locally cocartesian lifts of each leg gives you a locally cocartesian lift of the 'composite'. Using the characterization of locally cocartesian lifts from before, and the assumption that you're covering a thin 2-simplex, this becomes the statement that 'thin 2-simplices compose to thin 2-simplices' i.e. that 2-equivalences are closed under composition. (Again, really you check some filling condition and so on...).