Family Doesn't Want To Pay Nanny/housekeeper For 3 Weeks Of Live-in Work.

Hey everyone,
Looking for some advice or direction before we take this further.

My mom worked as a live-in nanny/babysitter for a family in Sunny Isles, Florida. Everything was verbal no written contract. When she was hired, the family said it would be $150/day (6 days a week) (at least that's the general pay in the area per day 150-200) . The 500/w rate was never mentioned she claims (6 days a week).
She lived in their home, slept in the same room as the 3-year-old on the 2nd level of the bunk bed, and basically worked full-time.

Payments were inconsistent sometimes Zelle, sometimes cash, often delayed. She has never raised a question as to where is my money and how much am i getting paid, hoping on the good faith of the family.
Later on , weeks in she realizes they're paying her $500/week(crazy, i know), which we now realize is much less than what was agreed. When she decided to leave, they didn’t pay her for the last 3 weeks (about $1,500) and now claim they “don’t owe her anything”
Not only they underpaid her drastically, they rejected to pay her for the hard worked weeks.

We’re wondering:

1. Without a written contract, can a verbal agreement + Zelle/cash records hold up in small-claims court or a wage-theft claim?
2. Do text messages, call logs, and her living there (neighbors, doorman as witnesses) count as proof of employment?
3. Is the Miami-Dade wage-theft program the right place to file, or should we go directly to small-claims court?
4. Could they argue she was “part of the family” or “helping voluntarily” since she lived there?
5. Would Florida minimum-wage protections still apply and what are our chances with pushing this further
6. What would be our next steps?

We’re not trying to start a fight just want her to get paid fairly (she just wants to be paid for the crazy amounts of hours of the work she already did.
Any lawyers, HR, or people who’ve handled similar domestic work cases in Florida what would you do in our shoes? We want to send a last letter of requesting the amount for 3 weeks otherwise continue this in court.

Thanks in advance.

Location: Miami,FL