What is a relatively painless way to move to a CMS without re-writing all of the Perl scripts?

One has nothing to do with the other -- a CMS need not touch the rest of the webserver, or any existing scripts

Are there script converters that will facilitate such a move?

No; Actually maybe; Depends on what the scripts do -- if they're just fetching content stuff from a database , you can export the stuff, and import into new cms database -- maybe then convert from markdown to new-cms-format-if-it-doesnt-come-with-markdown-parser

Which CMS is appropriate given this starting point

The starting point is generically vague -- its no different than starting from zero -- so pick one, pick two, doesn't matter

Is it required for the CMS to be Perl-based, or does this even matter?

Do you require the CMS to be perl based? I don't think it matters, it matters if the CMS has the features you want -- the less work you have to do the less work you have to do, its like less work

i.e. is there a way to run Perl scripts from within Joomla?

Is Joomla a webserver? Webservers can run any program, and there is no reason for two program to interfere with each other -- put them in different rooms (paths) and they'll live happily on the same webserver