This is a very interesting and tricky question. My best bet would be to try organic acids (formic acid, acetic acid). In my experience mixtures of these two acids are good enough solvents of PCL, even in presence of water: Article Polycaprolactone Electrospun Fiber Mats Prepared Using Benig...
In my experience, these solvents work worse for PLGA and PLA. Perhaps this will give you enough working time to combine the two components...
Otherwise, may I suggest looking into melt-based techniques? If I am not mistaken the melting point of PCL is significantly lower than the one of PLGA (60 °C against 250 °C or so). It could be a viable alternative.
I wanted to try the melt-based technique. However, the PCL is very viscous when melted and it is very difficult to mix it homogeneously with something.