I have a feeling your customers' concern has less to do with Perl per se, but perhaps more with the fact that you use your own homegrown custom CMS.
From a business continuity point of view, such is called a "single point of failure".
Coding a custom CMS in any other language or even making a site within the framework of any of the more popular applications, will not alleviate this fear: you will still be the only one who knows what makes the custom CMS tick.
It is not because anyone with two connected braincells codes PHP sites that they will be able to maintain some highly evolved website (even one written in PHP) with many bespoke modules and functions.
In essence, the decision your customers must take is: do they want a run-of-the-mill website straight out of one of the popular site building applications or do they want to have a site specifically tailored to suit their needs? And depending on your customer's answers, you will have to (re)position your business.
A program should be light and agile, its subroutines connected like a string of pearls. The spirit and intent of the program should be retained throughout. There should be neither too little or too much, neither needless loops nor useless variables, neither lack of structure nor overwhelming rigidity." - The Tao of Programming, 4.1 - Geoffrey James