$\operatorname{SL}_2(\mathbb R)$ Casson invariant?
Boyer and Nicas defined an SL(2,C) Casson invariant. The idea is to just ignore the noncompact components of intersection and you get a well defined invariant. My guess is their proof carries over verbatum to SL(2,R)
Dennis Johnson defined a geometric casson invariant. He never published it, but it is the sum over the irreducible representations of the Reidemeister torsion of the complex corresponding to cohomology of the manifold M with coefficients in ad of the representation when it is defined, and zero otherwise. He called it a geometric casson's invariant because he arrived at the torsion by computing the "angle" between the chaaracter varieties of two handlebodies from a Heegaard splitting of the manifold, inside the character variety of the splitting surface.
A 2020 arxiv posting of Nosaka (An $SL_2(\mathbb{R})$-Casson invariant and Reidemeister torsions) defines an $SL(2,\mathbb{R})$ Casson invariant. As Charlie's answer suggests, the approach is inspired by Johnson's unpublished work.