Taiwan Visa Overstay without passport

Hey guys, Im currently in Taipei and a few weeks ago I lost my passport.
I did everything that has to be done I guess.
-reporting it to the police
-applying for a new passport
-going to the Taiwanese NIA (National Immigration Agency)

But since my flight goes in just less than 2 weeks and my German diplomacy told me that it takes too long to get a new passport, I applied for a travel card to return. It is some kind of document that allows me only to return to my country, not to travel and it is also not really accepted by many countries since it is more of a national paper, not international.
My problem is that I will overstay my visa for ~30 hours… Actually the plan was to refresh the visa for an other 90 days, but for that you need a real passport.
I can also not travel to Hong Kong for 1 day, to restart the 90 days, since thats not what I can do with this paper.
The officials here all tell me not to overstay, never!! But there is not really much I can do about it.

So my question is: Do you think I can leave the country if I overstayed but only have like a national accepted return card? I do have the official police report for my loss, just like my certificate that I was studying here and also a copy of my lost passport.

Thanks in advance
Best regards

you went to all those government departments to report missing passport and never asked them this question?
wake up in the morning and go to one or more of them. Get answer in writing if possible.

I did kind of mention it, but didnt expect them to give me a real advice.
All they always said is do not overstay! Cant really ask them then “okay I will overstay what are my consequences”
But ye, I’ll go back to the NIA tomorrow and ask one more time

keep letter from your government stating the time it will take and show them. after you get passport immediately go again to show them, and see what they say. others may chime, but this is what i would do.