Dual citizen travelling within Schengen

I'm a citizen of both Egypt and the US and am currently studying in Italy on a permesso di soggiorno linked to my Egyptian passport.
My PDS expires near the end of September and I have already applied for a new one and gotten the receipt. I know you can only use this receipt to travel to and from your home country back to Italy and not within the EU, assuming your passport doesn't allow 90-day visa-free travel within Europe; which my Egyptian passport doesn't. However, my US passport does. Am I allowed to use it to travel to, say, Germany, considering all my entry/exit stamps are on my Egyptian passport and there's virtually no proof on my US passport that I entered/live in Italy.
Would be very helpful if anyone who has been in a similar situation can share their experience. Thanks!

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