I’m sorry for your loss at such a young age.
If the immediate purpose of your trying to get a citizenship is the coming 3-month stay, I think a visa exempt entry by your US passport could be an option. You can stay here for 90 days without a visa, and you need a visa run to stay for more than 90 days. Or, if your ralatives are here, you can apply for a visitor visa too, with their corporation. A visitor visa may be extendable once.
You can take short term classes of language, art, culture, etc, without a visa or TARC. Though you cannot enroll in regular schools，such as university, without a visa, it can be a reason to get your TARC.
I think most of volunteers may be allowed too, but this need to be checked.
If you understand Chinese, here is the instruction to apply for TARC.
https://www.immigration.gov.tw/ct_cert.asp?xItem=1088341&ctNode=32596&mp=1
僑居國外之臺灣地區無戶籍國民申請在臺灣地區居留送件須知
If your mom’s parents still live in Taiwan, you can get TARC based on your relationship with them.