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I am an asylee in the United States. I have a Green Card, and my primary passport which was issued by my country of origin. I am not allowed to go back to my home city and then return to the United States.

But, due to an emergency, I have to go there. So, I decided to use a third country to go to from the US, stay there for a while, and then go to my home city then.

Can I use the re-entry permit that gets stamped while departing the US, have my passport stamped in the third country instead of the re-entry permit? I would enter my home country with the passport, and return to the US by the same route, only by showing my re-entry permit and not the passport as if I have never been to my home country?

You should seek competent legal advice from someone with experience handling asylum cases, which may include local immigrant legal-aid organizations.

Even with permanent residency, traveling back home or even using the passport of the country you are claiming protection from can put your status at risk:

If your travel abroad suggests that you no longer need the protection
of the U.S., your status as a refugee or asylee may be terminated. If
you return to the country where you experienced past persecution or
claim a fear of future persecution, you may be required, upon your
return to the U.S., to explain your travel to that country to avoid
losing your asylee or refugee status.

A person granted permanent residence based on a grant of asylum is
still subject to the possible consequences of returning to the country
of claimed persecution. An person's asylum status may be terminated
even if the individual has already become a lawful permanent resident.
In some limited circumstances, you may be able to return to the
country you fear if your stay is of a short duration and you can
demonstrate that your return to that particular country was due to
compelling reasons.

Here's a useful blog post by an immigration lawyer who specializes in asylum matters: You Can Go Home Again (Sort of): Visiting Your Home Country After a Grant of Asylum. He writes on the ways such a trip could jeopardize your immigration status in the US, both now and if you later wish to apply for US citizenship and suggests collecting evidence about the nature of the emergency and consulting an attorney before you travel.

This trip puts your ability to return to and live in the US at risk. That's too important to rely on advice from strangers on the internet. Please get professional advice.

Don't do anything stupid ,please excuse my language, but this is not a advice it is a warning!!!!! any one has AS6 on their US green card ,don't try to go back to your country ,you will eventually lose ur stastuslose ur status ,CBP might let you in airport , but you will get a letter later from immigration court order before an immigration judge ,first you lied to us government, this is very common in Chinese case ,they will claim asylum, and after they will spy for communist party not all the Chiese are same but there are a lot. I work at uscis filed office ,so my friendly warning is wait until u became us citizen. We recently deported a guy from china who went to back china via France turns out he claimed chirsitin on his asylum but he was not ,so not only he went back to his country he lied on his asylum status.