As far as I can tell, most of the concrete models considered in (rigorous) QFT on curved spacetime are either free or perturbative. In fact the only construction of an interacting QFT on curved spacetime that I am aware of is by Figari, Höegh-Krohn and Nappi (PDF), developed further by Barata, Jäkel and Mund, for $P(\phi)_2$ theories on de Sitter space. This work uses the Euclidean approach via a novel kind of OS reconstruction.
In generic globally hyperbolic spacetimes, Euclidean methods no longer work. Given their usefulness, this is quite a blow to the constructive programme and I assume some new ideas will be needed. (There's a very interesting article about the unitary (un)implementability of dynamics for QFT on curved spacetime that indicates how some very basic concepts seem to break down for such theories.) There is some recent AQFT work on constructing the C*-algebras of interacting fields, I am also vaguely aware that Wald and Hollands suggested using OPEs, but it is not clear to me how successful these ideas have been or will be in producing actual models.
My question then is for those more in the know: are there at present any ideas in the community for constructing say $\phi^4$ theory on a general globally hyperbolic $2D$ spacetime, i.e. when Euclidean methods break down (besides the C* and OPE frameworks already mentioned)?
EDIT: Here is a reason to expect such a thing to be possible at least sometimes. The main problems in constructive QFT come from controlling the UV and IR behaviour of the theory. On the one hand, the UV behaviour of a theory on curved spacetime should really be very similar to flat spacetime since every space looks locally flat at small scales (perhaps if we forget about singularities for the moment), while we could try to sidestep the IR problems by working on spacetimes with compact Cauchy slices (this isn't completely general, but it's a start). On the other hand, the historically first constructions of $\phi^4$ on Minkowski space by Glimm and Jaffe proceeded directly in real time without Euclidean methods (see e.g. Summers' review). Of course Minkowski space does have a lot of symmetry, so I don't really know what to make of this argument.