I think the immigration strategy can just be for your mother depending on how difficult your mom’s case is for establishing her own hukou (did her parent(s) already have hukou when your mom were born)?
Because once your mom’s hukou situation is resolved, Both you and your brother and kids are eligible to apply for a TARC under AF353 where all of you stay 335 days out of the year to convert to HHR.
If it were me, I would try working with one of these first and maybe get a second opinion from a lawyer if they can’t help. Please be aware that Taiwan basically does not have immigration lawyers per se. Much of the filing and application type work that an immigration lawyer does in the US is better handled at less cost by one of these licensed professionals. That said, lawyers can be extermely helpful with difficult cases and may be able to see solutions or strategies that a licensed immigration consultatant cannot. Lawyers are much more expensive and will have to charge you for the time they spend coming up to speed on the law in this area. That seems very unreasonable to many but I disagree. YMMV.
See Articles 55-57 of the Immigration Act for information about non-lawyer licensed immigration professionals. They are bonded. You should check the licensing status of any professional in Taiwan before you hand over money.