The problem is unclear. First you say "I cannot get the output of the internal
make to be present in both"; then you try something and "this redirects make output to the output file". I know how tee works: if something gets to the output file then it must get to the stdout of tee. So you can"get the output of the internal make to be present in both", right? Now the problem is with vague"undesired verbose logging". What is it exactly? Where does it come from? The question begs for mcve. Please [edit] and add one.