Issue moving family to Italy from UK

Hello,

I am a French national, married to a South African who has now the British nationality. We got married in the UK and have two young children. I got offered a job in Bologna, starting in April. Regarding immigration, the relocation service of the company who hired me said that it is not possible for my wife and children to come with me at the same time as me. She says we need to register our marriage certificate and the kids' birth certificate to the French authorities, mentioning a family cohesion scheme. The issue we have is that to register that in France, we need a legalised or apostilled copy of my wife's birth certificate. We called the South African embassy in London and they told us it could take up to a year or more to obtain this document. Once we have this, we can send our application to the French Authorities, to register the marriage certificate, and it takes another 6 months. Then, we can register the kids birth certificate, which takes another 2 months.

So this means I could go to Italy in April, leaving my wife and children in the UK for 18 months before they can join me, which is not reasonable nor acceptable.

We have looked online and we actually don't understand why we have to do what I described above.
According to some webiste on registering non-EU family members in another EU country, it seems that we can all move together to Italy at the same time. Once we move and find a permanent accommodation, within a few weeks, my wife and kids should be able to register to obtain a residence card. "The authorities should make their decision to issue a residence card or not within 6 months. If they do not do so, you can call on our assistance services. In any case, your non-EU family members cannot be expelled if their visa expired while their application is being processed." As they would be one a 3 month "visa", and that we will be in the process of registering, it clearly says that they cannot be expelled while the application is being processed.

Could someone provide clarification on what the process is exactly so we can all move together at the same time? If we can't do that and have to wait for 10, 12,18 months or more, obviously this is a serious issue and goes against this so called family cohesion scheme.

Thank you!