In theory, there is no requirement to register anything with the consulate and you don't need your father to file anything for you. Doing that immediately after birth makes a lot of things easier down the line but does not per se change your citizenship.
You need to present good evidence of all the relevant facts and that might obviously be more difficult without his help but that's all there is to it. In fact, your father's testimony does not carry much weight, a friend of mine had a lot of trouble registering her daughter's birth because of some bureaucratic issue, what you really need is proper documentation, nothing more, nothing less.
Specifically, you need to establish that:
- Your father is your father and you are his son. Legally speaking, your local birth certificate should be good for that, even if I personally know some French consulates making difficulties to accept some local documents – not a birth certificate in my case – for no good reason.
- Your father was a French citizen at the time of your birth. His passport and birth certificate, while not incontrovertible proof of citizenship, are relevant here. And if his birth was registered with the Service central d'état civil (which, again, is not mandatory), as his son, you can even get an extract of his French birth certificate without his help. But you need to be sure that he was French when you were born. The phrasing of your question (“reclaim his citizenship”) initially threw me off but your last comments suggest he was already French then. Whatever the case may be, that's what you need to establish, his becoming French later on would be no good for you.
- If relevant (if both of you have lived abroad for more than 50 years), that your father and/or yourself have had possession d'état at some point in the last 50 years (see, e.g., Grandfather served in the French army for 4.5 years: Can I get French nationality? for more details on what that means). Your father's passport is helpful for that (with the caveat that you only have a copy and that I have no idea how all this works with Djibouti, which hasn't even been independent for 50 years).