I'm still shocked there appears to be no rechargable option!

It is a weird uncommon battery size that the manufacturer custom made for their product. Also replaceable rechargeables only make sense (to the manufacturer) over a limited range of sizes. Very small devices (like stuff that is worn) and very large devices (like cars) usually will not have replaceable rechargeable batteries.

I've heard you can splice open the outer wrapper and it's actually a couple regular sized button cell batteries under... but I haven't investigated that yet.

Yup. From the very page you linked: "One assembly made with two CR-1/3N cells shrink wrapped together." And those button cells cost about $1 each normally.