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I am sorry to hear about your situation. As a lawful permanent resident (i.e. greencard holder), you are expected to reside in the USA. If you leave USA for 6 or more months, USCIS/ICE can assume that you have abandoned your greencard and can have it VOIDED right then and there at the entry point, and not let you in. You would lose your LPR status.
Yes, you can bring a packet showing ties to the USA (permanent residence, bank account, etc), but this is subjective and there is no guarantee that USCIS/ICE will see it as proof enough that you have not abandoned your residence. In the end, this is THEIR decision to make.
You are telling me that you have come home every summer, which means you spend 9 months abroad and 3 months in the USA every year. You are lucky that you have not lost your LPR status already and only received a warning!
The I-131 takes about 90 to 160 days to get processed. There is no formal expediting process. If you do not have time to wait, it is suggested to at least file it and then make an appointment with INFOPASS (here) to explain the situation to a USCIS officer, who may expedite it via informal channels, and (in rare cases) can even grant an I-131 then and there.
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