They did that back in the day, with the LS-120 and -240 "SuperDisk" and I think a couple of similar but even less well known systems. They used a secondary head tracking servo information on the medium in order to achieve tracking precision beyond what was available with the standard floppy design, and thus (no doubt along with reformulated media) write more information in the same area. They were back-compatible with regular 3.5" floppies, but they were also slow, flaky, expensive, and introduced after Zip disks but not long enough before writable CD media, and so they failed to compete.
If I were to do it today, I'd probably think in terms of USB, with an annular PCB loaded with flash chips in place of the disk medium, and a set of pads exposed by the slider to mate with pogo pins on the drive's head sled. (The PCB would be fixed in the disk case, with the central "cookie" remaining in place to mate with the drive spindle.) You'd almost certainly need to modify or replace the drive firmware, which in its default state would likely find this all very confusing, so it could spin the "disk" but avoid moving the heads (and thus interrupting connections and maybe also doing mechanical damage) and thus give the added realism of motor noises that stock USB drives just can't match. For extra bonus points (and a saleable product!) replace the firmware (and the onboard controller hardware) with something that can talk USB to the inserted "disk", and the native protocol out the board-edge connector to the computer in which the drive is installed - and then you have one of those "multidisk" devices for vintage machines, but one that actually uses a disk, and you can make a mint from retro nerds like me who enjoy old machines but are too busy to deal with them in their fully stock configuration. [1]
I'm the wrong kind of engineer for a project like this, but I wish I weren't, because it sounds like a lot of fun.
[1] Well, not me personally, because the only machine I still have with a 3.5" drive is a Toshiba Tecra from 1995 or so that uses all the exuberantly nonstandard interfaces one would expect of such a year. The internal floppy gave up the ghost ages ago, but an external one and an Ethernet PC card keep it able to talk to the world when it wants to. Heaven forfend its hard drive ever fail, though...
edits: correcting misrecalled SuperDisk technical details