- Be a partner and publish your paid app there. It then will be paid.
- Submit a DMCA notice: "9.3 DMCA Notices. If Aptoide receives a Notice according to the US Digital Millennium Copyright Act (DMCA), Aptoide will immediately remove the Product in question. Such notices shall be directed to Aptoide’s DMCA Agent at abuse.report@aptoide.com." They will then unlist it.
DMCA is a nuisance. You have to be big enough to dedicate resources to filing and tracking takedowns, as well as be able to take legal action anywhere in the world for sites that don't respond. It's very easy to clone a pirate site. It's much harder to play whack-a-mole to take one down.
At work, we've used MarkMonitor to try to solve similar things, which is sometimes useful and sometimes results in our secondary apps getting taken off Google Play.
I suppose you could send them an Electronic Commerce Directive take-down notice. Probably doesn't hurt them as such, but should get your app removed.
EDIT: The store seems to be somewhat decentralized, with stores being similar to APT repositories. So it may be difficult to target Aptoide directly unless there's a centralized list of stores, in which case I imagine you can get the user's store struck from it. Otherwise you may need to contact the store's hosting provider - which indeed could be a bit of a cat and mouse game.
The FOSS store f-droid also allows to add custom apt-like repositories, which for example CopperheadOS used to add a non-sanctioned patched version of the "signal" app. Moxie doesnt want forks of signal that are also using signal's servers. CopperheadOS did it anyway because signal is not compatible without Google Apps installed (removing google cloud messaging because CopperHeadOS doesnt include Google play services).
F-Droid is a fork of the original Aptoide project, it seems.
I've already had to contact a few devs to be able to use paid apps because they thought that root = pirate.