I am 100% sure that although the signing bonus and transfer money are different expense items in an accounting sense, they are both still taken into account by the FFP. But Real can afford such expenses because its income exceeds expenses, they have not made major purchases for a long time (before Bellingham) and have accumulated a good financial reserve.
In addition, I am sure that Mbape will receive his signing bonus not just once, but as the contract is fulfilled, that is, this payment will be spread over several years.
And there are many more bonuses that Mbappe can get outside the formal context. Real Madrid doesn't always buy players often, but once they do, all the teams will be amazed to see them. It took just one Mbappe signing to become the biggest transfer of the season. Regarding managing income and expenses, Real Madrid has a clear structure so that it does not waste money carelessly. On paper, Mbappe is a free transfer and this is the genius of Florentino Perez that no other club president in Europe has.
Looking at the other side, the rivals are experiencing reverse conditions, struggling to recover from bad finances, difficulty maximizing player investment, lack of facilities and behind this it is all the result of the club's financial management not being neatly organized. In the current era of football, stable finances will create an ideal squad. Real Madrid is a reflection of the situation of a club that can do everything, Florentino Perez started from zero and now just has to reap the results.