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.
This is well known, but formulated in a slightly different way.
Recall that a Frobenius category is an exact category which has enough injectives as well as enough projectives, and such that an object is projective if and only if it is injective (injectivity (resp. projectivity) is defined with respect to inflations (resp. deflations)).
If you forget about the existence of (finite) limits and colimits, any Frobenius category satisfies all the axioms of a Quillen model category: the cofibrations (resp. fibrations) are the inflations (resp. deflations), while the weak equivalences are the maps which factor through some projective-injective object. Therefore, any Frobenius category which has finite limits as well as finite colimits is a (stable) closed model category in the sense of Quillen.
Now, given an additive category $A$, the category of chain complexes $C^\sharp(A)$ (where $\sharp=\varnothing$ for unbounded chain complexes, $\sharp=b$ for bounded chain complexes, etc) is a Frobenius category: inflations (resp. deflations) are the degreewise split monomorphisms (resp. split epimorphisms), and projective-injective objects are the contractible chain complexes. If $A$ has finite limits as well as finite colimits, this shows that the category $C^\sharp(A)$ is a stable closed model category whose cofibrations (fibrations) are the degreewise split monomorphisms (resp. epimorphisms), and whose weak equivalences are the chain homotopy equivalences.
Best Answer
Here is an example that surprised me at some time in the past. Bisson and Tsemo introduce a nontrivial model structure on the topos of directed graphs. Here a directed graph is simply a $4$-tuple $(V,E,s,t)$ where an arc $e \in E$ starts at $s(e) \in V$ and ends at $t(e) \in V$. Fibrations are maps that induce a surjection on the set of outgoing arcs of each vertex, cofibrations are embeddings obtained by attaching a bunch of trees, and weak equivalences are maps that induce a bijection on the sets of cycles.
In this model structure fibrant objects are graphs without sinks and cofibrant objects are graphs with exactly one incoming arc for every vertex. Cofibrant replacement replaces a graph by the disjoint union of its cycles with the obvious morphism into the original graph.
We have a chain of inclusions of categories $A\to B \to C\to D$, where $D$ is the topos of directed graphs, $C$ is the full subcategory of $D$ consisting of all graphs with exactly one incoming arc for each vertex, $B$ is the full subcategory of $C$ consisting of all graphs with exactly one outgoing arc for each vertex, and $A$ is the full subcategory of $B$ consisting of all graphs such that $s=t$.
Each functor is a part of a Quillen adjunction and total left and right derived functors compute nontrivial information about graphs under consideration.
Two finite graphs are homotopy equivalent iff they are isospectral iff their zeta-functions coincide.