actually, I’m not sure what you mean in this part.
Iiuc, your mother can get an entry/exit permit and stay in Taiwan for 3 months, and she can extend it once without leaving taiwan. So, she should exit taiwan once every 6 months. I don’t think she needs a petition for this.
During her stay in Taiwan, she can work if she wants, because she doesn’t have any other nationality. She can also enroll to NHI through her employment.
If she stays in Taiwan more than 183 days per year for 7 years, she would be eligible to get a TARC. After she gets a TARC and stay in Taiwan for a year, she can get a household registration.
The above is case 4 in the following page of TECO in Manila.
I now understand that you said about case 3 in the page. Though the instruction says petition, maybe it should be applied by your mother with the national ID and transcript of active household registration of her sister or brother.
Some related articles and instruction.
Short-term residency of nationals without registered household in Taiwan area
Visa extension for nationals without registered household in Taiwan area