After a ten month internship in corporate finance from January to September, I was hired in October and finished my trial period at the end of December.
The firm may give bonuses after a successful internship and to people employed for a complete year. This happens once a year in February. Like many others, I was not offered a bonus for my time as an employee but was promised something for my internship.
After what seems to be a mix-up from the HR, my internship bonus was forgotten and the deadline is passed. My manager (new to the team) and the CFO offered a solution :
I am to say nothing about this and say (if asked) that I had nothing. On the other hand, they will find an alternative way to pay me a bonus.
I feel very unconfortable with this situation : as a moral principle, I feel I have to refuse this offer. But if I refuse, I must find the right words and a way to use this for the next negociation.
I have two days to solve this problem and would really enjoy your help!

What's the moral principle?
A religious leader once gave me a great test:
Say it if any 2 of the 3 are yes.
The manager and CFO want to avoid having HR lose face for their screwup, because rubbing HR's noses in the mess they made will just make it harder to resolve the problem. That means more of their time and energy, and they don't have a lot to spare.
So they propose a solution. Is it true that HR screwed up? Yes. Is it necessary to tell people that they screwed up? As I see it, not really, no. Morality is served by having you get what was promised in another way. Is it nice to say that HR screwed up? No, it isn't.
That's only 1 of 3. It's true to say, but not necessary, and not nice. Their solution passes the test.
Now, what if someone asks you?
Is it true to say that you got no bonus? No. Is it necessary? Well, if you think helping your manager and the CFO avoid a real drain on their time is necessary, then yes, it is. I'd suggest to you that their willingness to help you entitles them to this consideration. Is it nice to say "no, I got nothing"? Well, the other person may be in a similar situation, and won't feel bad. And it also avoids telling people that HR screwed up, which is true but not nice (or necessary, see above).
So yeah. Not true, but necessary, and nice, to say you got nothing. That passes too.
I'd take your CFO's offer, and know that morally, you're OK here.