Thinned Wallis-type prime products in residue classes modulo $2^m$

For odd primes $p$ we consider the factors $$A(p)=\frac{p-{\chi }_4(p)}{p+{\chi }_4(p)},{\chi }_4(p)=\{\begin{array}{ll}1,& p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4),\\ -1,& p\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4),\end{array}$$ and study products of $A(p)$ restricted to unions of residue classes modulo $2^m$. We give a simple criterion for the existence of a finite nonzero limit, prove a logarithmic asymptotic in the general case, and express the limiting constant in terms of Mertens-type constants in arithmetic progressions (hence in terms of Dirichlet $L$-values).