Actually Dist::Inkt is unlikely to be much help; it only does one job: produces a tarball suitable for uploading to CPAN from the project directory. (That includes building META.yml, META.json, Makefile.PL, etc.) Creating a new project directory from scratch is something outside its mandate.
However it has a companion project Dist::Inktly::Minty which does that; it should be fairly easy to subclass to do what you want. (A couple of subclasses are bundled with it.)
Update: as a general design note, if you do publish Module::Cooker... don't waste your time making it super-configurable with config files, etc. The end-users it is aimed at are Perl programmers; they are not opposed to writing Perl to get something done. So just make it easy to subclass - this is a far more powerful form of extensibility. Break everything up into small methods that can be overridden in subclasses; document them, the arguments they take and what (if anything) they're expected to return. Make sure that your command-line tools support a --cooker=Module::Cooker::MySubclass option to allow users to indicate hey want to use a particular subclass of Module::Cooker rather than the base Module::Cooker package.
use Moops; class Cow :rw { has name => (default => 'Ermintrude') }; say Cow->new->name
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