Why does Mars work with this organisation for secret research?

Martians are known for their excellence in science and research (stealth ships, the best navy, etc). Why wouldn't they use their own private companies, or perhaps government organisations

It is unclear exactly how much the MCR outsources on a daily basis. Just because MCR has state-of-the-art technology does not inherently prove that they made it all from scratch themselves.

On top of that, if your research is bound by moral limits (which MCR seems to be), you'll end up with less/slower progress on researching the protomolecule. Mao performs human experiments not because he wants to be evil, but because it is the fastest way for him to research the protomolecule and he cares little for the human lives lost.

So if MCR looks at their in-house expected research time, and Protogen claims to be able to do it much faster (for a reasonable cost), then it makes sense for MCR to outsource the work.
This doesn't even require the MCR to be complicit in Protogen's human rights violations if there is reasonable doubt whether they were aware of Protogen's intentions at the time of outsourcing the contract.

Why wouldn't they use their own private companies, or perhaps government organisations, rather than risking everything with an Earth-based company with an Earth owner (Jules-Pierre Mao)?

I'm not sure which season this is highlighted in but I think it is the earlier seasons, but note that Mao's company is not as Earth-based as you suggest it is. Mao is perfectly able to avoid Earth's scrutiny when it does come (somewhere during the show) and he's not bound by Earth laws as much as he theoretically should be.

His wealth has almost made him a nation state of his own (this is an analogy, not a literal spoiler as far as I'm aware), which makes him less of an Earth puppet than your question implies.