One particular comment from your question is concerning:
The job has a problem with retention due to our duties being absurdly
micromanaged by government overseers. (About 80% of new hires either
quit or are fired within two weeks.)
That means either the company isn't sending quality employees, or the government customers is not a good customer. Either way the situation can't continue for long. The management of the customer will soon notice the poor performance of the contract and will investigate. That could bring the contract to an end. It could also end the company if that contract is most of their business.
The rumored misinformation given to people performing background checks can be an issue. Many large companies outsource the task of confirming dates of employment for former employees. That service just provides dates of employment, maybe job title, and very little else.
But because "The owner is an individual person who handles all H.R. matters themselves." you can run into an issue where they see no problem either providing inaccurate information, or no information at all.
Leaving this company shouldn't be a problem, because most companies realize that your current boss has a vested interest in either embellishing their praise or minimizing their praise. You are correct the next transition will be harder, because the newest employer has no reason to doubt the comments from the company.
My advice would not to be overly concerned about it. Document what your can about your performance. Document your starting end ending dates from company supplied documents: the offer letter, the welcome packet, any exit documents. Pay stubs and tax forms are also helpful.
The background checks that I have participated in have always allowed potential employees to explain/prove missing information. A company that won't confirm dates is easy to overcome. Claiming you wouldn't be rehired implies either extremely bad performance or committing a fireable offense. You documentation should also be able to overcome that.