What you can prove is that if you take a $2$-complex $X$ with fundamental group $A\ast B$ and you take a finite subcomplex $K$ of a covering $X'$ of $X$ such that the fundamental group of $K$ maps under the covering to a subgroup of $A\ast B$ which is closed in the profinite topology, then the restriction of the covering map to $K$ can be extended to a finite sheeted covering.
An equivalent formulation in the language of Bass-Serre theory can be found in the paper of Ribes and Zalesskii PROFINITE TOPOLOGIES IN FREE PRODUCTS OF GROUPS