Oversea (PRC) Chinese staying in Taiwan for 2+ weeks?

I am going to study in Taiwan, and my girlfriend wants to come with me for the first few days/weeks. So we booked a flight together, me staying 6 months and she staying 18 days.
However, after buying the ticket we found out that she, holder of a PRC passport, is only allowed to stay there for 15 days (It says so here: immigration.gov.tw/ct_cert. … 32595&mp=1) and it seems like they only do extensions in emergencies.

Do you have any ideas how to solve this nicely? Things we have considered:

  • Changing the flight. - This option is obviously the safest, but it’s 400€, so quite expensive.
  • Ask for extension - 3 days extensions is not that much, however if we go there and they tell us “No” and she has to leave the country in X days, it will probably be even more expensive. TEPCO didn’t have any idea, wether they are going to extend this or how strict this is.
  • Leave Taiwan and enter again - Much of a hassle, because she would need another 台灣地區出入許可證 + time for flight + price of the flight.
  • Overstay. Haven’t found any experiences for that, but she probably wouldn’t do it unless there is a regulation that guarantees that she only has to pay some kind of fine and nothing else. (Like a black list or god knows what…)

Somewhere I read about different situations, if she lives abroad for more than 4 years (which she does, in an EU-Schengen state), but I couldn’t find anything that applies for our situation. I’m glad for any help or valuable tip.