What you asked
Fundamentally, you're asking:
How to [motivate] the rich to [raise] more children.
You also provide the following constraint:
By law every child gets equal share.
which allows us to ignore the possibility of children being disowned and disinherited.
But you also specified:
All those charity funds are just shams to pay less taxes.
here's a whole industry of accountants and lawyers to make wealthy pay less taxes.
So how do we motivate the wealthy to raise more children? By looking at what currently motivates them and structure raising children as providing a similar benefit. In this case, they are very clearly motivated by a desire to pay less taxes. Therefore,
Legislate tax breaks proportional to the number of kids raised.
Write the legislation so that the amount of money saved from taxes per child is greater than the amount of money spent to care for each child per tax year. This creates a net-tax break. Structure the legislation so that raising more kids returns a greater tax break. More importantly: do not cap it leaving it uncapped maximizes the generational distribution of wealth.
This creates a dangerous issue: exploitation
How does one protect the children from malicious upperclassmen who intends to exploit them for financial gain?
It would seem simple to legislate the following points:
1. Each child must be guaranteed a share of the inheritance.
2. Each child must be treated humanely (and this much be checked)
3. If the child dies or goes missing, formal investigations must be had.
But, we're talking about the supremely wealthy and their armies of lawyers and accountants. Obviously, if they'd like to maximize their tax cut, they will be motivated to adopt a massive number of children. In the short term, this benefits them; however, in the long term, they may wish to keep their wealth and not distribute it after their death. Similarly, they may wish to keep the children at a manageable age to reduce costs and psychological overhead. Anecdotal example: rich teens invokes far more legal fees than rich 10 year olds. Or, they could sell the kids (ex: females) when they reach a certain age and make a tidy profit. Therefore, they may attempt to trade or remove kids from their parentage, be it adoption to another wealthy family, or 'mysterious disappearances' (read: murder, human-trafficking, slavery, prostitution). Without heavy government oversight, it would be simple for the obscenely wealthy and obscenely malicious to exploit lower-class adopters for financial gain. Similarly, one can see them forming the equivalent of a foster home.
In essence, you'd be shifting the foster-care system to the wealthy; but, like the current foster care system, incredible oversight will be required to prevent exploitation.