I copied the "MATLAB Compiler Runtime" folder and contents from the older PC that ran the .exe to this newer one and added a line in the PATH to the runtime\win64 folder. That was sufficient for the .exe to run successfully without actually having the MCR installer.
I had tried this previous to asking the question here but I still got an error. Turned out that the application was also dependent on Microsoft Visual Studio 2005 Redistributable.