Bounded Ramsey's theorem for triples in computability theory

We study a restriction of Ramsey’s theorem for 2-coloring of triples, in which homogeneous sets for color~1 are of bounded size (${\mathsf{B}\mathsf{R}\mathsf{T}}_2^3$). We prove that the computational content of this statement is very close to Ramsey’s theorem for pairs (${\mathsf{R}\mathsf{T}}_2^2)$, in that it satisfies the same known computability-theoretic upper bounds, but that ${\mathsf{B}\mathsf{R}\mathsf{T}}_2^3$ is not computably-reducible to ${\mathsf{R}\mathsf{T}}_2^2$, even when allowing multiple applications of ${\mathsf{R}\mathsf{T}}_2^2$.