A few years ago I used a tool from MySQL to migrate from Access. It had a nice GUI, and it worked really well, MySQL Migration Toolkit. The documentation states that this is now EOL, but suggests checking out MySQL Workbench. IIRC you have experience developing using CGI::Application, from conversation and reading your posts here I'm sure you're more than capable of the doing this migration. At work I use jQuery and jQuery UI extensively for our UIs.

Regarding no one giving a second though about a currently operational system, I'd disagree that this is how it should be. As you are finding now there are costs involved in maintaining this, which are perhaps an issue. In the past you've run out of disk space a couple of times over the years IIRC. While I agree, end users should not care, someone should be wondering where systems and services are going in future.

To this end, since you'd be migrating the legacy systems into a new one you will have to design, build and maintain, is it time to ask for a raise ;)