By carrying out your suggested test, I identified the two modules as process.pm and changenotify.pm.
I looked in both modules and could see both had INFINITE in the EXPORT. However, I could not recognize where either defined what INFINITE meant. Therefore I suspect that there is some incompatibility between these two.
You also said “If the other place that imports it doesn't have a way to prevent it too, then a work-around would be to use it inside a dummy package. It is only used in one line, so delegate that line to a function in that inner package, which you call from the main code.”
Although I have just Perl for some years it would help if you could give some more details.I guess I should know(!) but I simply do not understand what you mean particularly with the ‘so delegate’ part of it.