So, it's fine with Perl which ships with your distribution, but not ActiveState Perl on the same machine. My initial suspicion is that this is due to how the Tk has been built by ActiveState. README:

By default Perl/Tk makes use of client side fonts via freetype2,
fontconfig and Xft on Unix systems. This gives anti-aliased fonts for
regular text (if you have TrueType or Type1 fonts and they are in your
fontconfig config file). See also http://fontconfig.org .
To disable this feature run the Makefile.PL
with "perl Makefile.PL XFT=0".
[download]

Perhaps you need to do some digging in order to find out the differences between distributions. Out of curiosity is there a specific reason you're using ActiveState Perl on Linux?

The idea was to develop under Windows using ActivePerl and just cross-package the final result to Linux without any major effort. But after diving into it, the reality looks somehow different.
The answer may be: Using ActivePerl with PerlApp under Windows and distribution-supplied Perl with PAR::Packer (pp) under Linux.
Thanks for your help.

I agree with your proposed solution, if you're packaging for different platforms, while requiring some more work on your part I think the results will be more favorable.