What are the consequences of registering at the address of ex-partner?

Questioner

After our divorce we decided to move to another municipality. I was able to find a house there but my ex-partner has to rely on a rental house and the waiting list for social housing is >7 years. Because our children (joint parenting) are starting their new school year there my ex plans to temporarily live at a holiday park nearby. Registration as a residential address is not possible there. You have to be registered somewhere. She is a self-employed person. Would it be better for her to register with her parents (>80 km away, no assistance, no rent) or at my new home address? And what are the risks of trouble with which government agency(ies), who do not care about reality and the (as much as possible) good faith actions of citizens.

Lawyer

Address investigation can take place on one's own initiative or by a tip from the municipality or other agency and certain allowances and such that are normally received for children must then be repaid, moreover, there is then intent. When registering at the address, you remain fiscal partners, your address also remains a multi-person household for municipal taxes.