I'm not aware that the model category you want has been constructed. But it seems like an interesting question. You should ask Julie Bergner if she has thought of anything along these lines.
I don't have an answer, but I will think out loud for a bit.
(I'll be a little vague by what I mean by "space", but I probably mean "simplicial sets" here.)
I will assume that your property 3 should say: "The weak equivalences between fibrant objects (i.e., between Segal spaces) are the DK-equivalences." I would also like to throw in an additional property:
4. The trivial fibrations between Segal spaces are maps $f:X\to Y$ which are DK-equivalences, Reedy fibrations, and such that the induced map $f_0:X_0\to Y_0$ on $0$-spaces is surjective. (Note: since $f$ is a Reedy fibration, $f_0$ is a fibration of spaces.)
Yes, this comes out of thin air ... but it's modelled on the trivial fibrations in the "folk model structure" on Cat. You could go further, and posit that fibrations between Segal spaces are Reedy fibrations such that $f_0$ is surjective.
Given a space $U$, let $cU$ denote the "$0$-coskeleton" simplical space, with $(cU)_n=U^{\times (n+1)}$. If $U$ is a fibrant space, then $cU$ is Reedy fibrant; if $U\to V$ is a fibration, $cU\to cV$ is a Reedy fibration. Furthermore, $cU$ clearly satisfies the Segal condition.
Thus, if $g:U\to V$ is a surjective fibration of spaces, $cg: cU\to cV$ should be a trivial fibration in our model category, according to 4.
The functor $c$ is right adjoint to $X\mapsto X_0$: that is, maps of simplicial spaces $X\to cU$ are naturally the same as maps $X_0\to U$ of spaces.
Putting all this together, we discover that, if such a model category exists, a cofibration $f: A\to B$ should have the following properties: the map $f_0 : A_0\to B_0$ is a cofibration of spaces, and $B_0=B_0'\amalg B_0''$ so that $f_0$ restricts to a weak equivalence $A_0\to B_0'$, and such that $B_0''$ is homotopy discrete (i.e., has the weak homotopy type of a discrete space).
In particular, a necessary condition for $B$ to be cofibrant is that $B_0$ is homotopy discrete.
This is a pretty restrictive condition on cofibrations, but it does not seem impossible. If there actually was a model category with all these properties, it appears that the class of fibrant-and-cofibrant objects would be what you might call the quasi Segal categories. These are the Segal spaces $X$ such that $X_0$ is homotopy discrete. Cofibrant replacement of a Segal category would give a DK-equivalent quasi-Segal category.
That would be a pleasing outcome, and probably along the lines of what you're looking for.
As you say, a major use of $(\infty,2)$-categories is for organising $(\infty,1)$-categories and similar objects (stable $\infty$-categories, $\infty$-topoi, enriched $\infty$-categories, $\infty$-operads...). The importance of the non-invertible 2-cells is the same as in classical $(2,2)$-category theory: they provide natural internal notions of adjunction, base change mappings, lax functor, lax monoidal functor, Kan extension, and so on. An $(\infty,2)$-category can be used to organise collections of these structures and keep track of coherences between them.
A good illustration of the utility of this is the notion of a "six-functor formalism," which Gaitsgory and Rozenblyum (https://bookstore.ams.org/surv-221/) argued is best captured by a certain symmetric monoidal $(\infty,2)$-functor on an $(\infty,2)$-category of correspondences between derived stacks. We definitely need $\infty$ here because the value of such a functor would be something like the derived $\infty$-category of quasi-coherent or constructible sheaves, and the source may also include some derived/higher objects. We definitely need $2$ because the 2-cells of the category of correspondences encode all kinds of coherences between the six functors (for example, base change 2-cells and the higher associativity of compositions of 2-d grids of base change squares). Even if one ultimately only cares about constructing functors out of correspondences on a 1-category, in practice one still needs its universal property among $(\infty,2)$-categories.
See also my paper https://arxiv.org/abs/2005.10496 for a slightly different take to Gaitsgory-Rozenblyum's.
Best Answer
A category object internal to simplicial sets is the same as a Segal space in which the Segal conditions hold on the nose instead of merely up to weak equivalence. In other words, a category is something whose nerve has unique horn fillers instead of merely contractible spaces of fillers.
The above category objects generate a full sub-(relative category) of Rezk's relative category of complete Segal spaces. As I explain below, Barwick and Kan's work proves that the inclusion of this sub-(relative category) induces an equivalence of homotopy theories.
Barwick and Kan construct a nerve functor $N$ from small relative categories to simplicial spaces. The key point is that anything in the image of this nerve is a category object in the above sense.
Their nerve functor $N$ has a left adjoint $K$, but they also consider a second functor $M$ from simplicial spaces to relative categories. The functors $M$ and $N$ are inverse equivalences of homotopy theories in the sense that there is a zigzag of natural weak equivalences $$NMX \rightarrow NKX \leftarrow X$$ for any simplicial space $X$, and a natural weak equivalence $$MNY \rightarrow Y$$ for any relative category $Y$.
If one restricts the domains of $K$ and $M$ to consist only of category objects, the above natural weak equivalences remain intact. Thus the Barwick+Kan homotopy theory of relative categories is equivalent to the theory of category objects in simplicial spaces.