AFAIK there is no convention so it's a design decision for you to take.
You could look at the Linux Filesystem Hierarchy Standard (FHS)
for inspiration. According to the FHS, /mnt and its subdirectories "is provided so that the system administrator may temporarily mount a filesystem
If you don't regard your user as a system administrator you may prefer ~/mnt/Y. The FHS says nothing about subdirectories of users' home directories so no guidance there.
These considerations alone suggest that a .BAT interpreter would at best be a kludge, as is further suggested by real world .BAT files being largely composed of commands which do not exist on Linux.
Perhaps you could write a Linux language script tailored to translate the specific contents of your logon scripts.