The mesh network should be made out of common hardware in order to be viable. I'd suggest phones but those devices are owned before they've even left the factory.
"Star Wars Episode 10: The one that's not fiction."
On the other hand, if you want to demonstrate that you have anti-satellite capability it's probably a better idea to shoot down a corporate satellite than a military one. The Soviet Union shot down Korean Air Lines Flight 007 and it didn't start a war, after all.
Cryptocurrencies might be a problem in this plan, and satellite internet access itself might become a currency (since unlike cryptocurrencies, this one both has almost an intrinsic value and provides its own infrastructure that's very hard to block, where as cryptos rely on external sources of Internet access).
It also depends - drugs have consistently won the war on drugs despite being a physical product that needs a local supply chain and various anti-money-laundering and banking/finance regulations that should make it hard to fund the operation. Satellite internet access is likely to be even easier as it doesn't rely on a physical product (if we reach this stage there's going to be clandestine satellite terminals built locally, so blocking shipments of the real thing isn't going to cut it).
The only solution, apart from North Korea-levels of isolation (and even then, NK has the advantage of their population being isolated & indoctrinated since birth, something most other countries won't achieve even if they turned authoritarian overnight) would be detection followed by harsh punishment, but this has the downside of not only wasting the disclosure of detection capabilities (that are useful to the military) but also outsourcing the R&D of evading such capabilities into the open which enemies will no doubt pick up on too and use against you in a conflict.
So it can (at least eventually) function without 'regular internet', although I would still be hesitant to call it a viable infrastructure choice if the goal is to get around government control, simply from how much SpaceX have to appease the government to do anything space related.