It’s definitely tricky to navigate these permit rules in Croatia.
In summary, to rent a boat under the Croatian flag, you would need a permit, but there’s no reason for them to ask for one under the French flag since French law doesn’t require it.
I don’t have one, and I’ve rented twice in Croatia. The first time, I showed a fake permit, and a course certificate also works. The second time, they didn’t ask for anything. However, I do have a CRR.
In 2016, I bought my boat in Croatia and registered it under the French flag before leaving Croatia.
I completed the formalities for exporting a Croatian boat and importing a French boat with the authorities, and no permit was asked for.
With your own boat, you must strictly adhere to entry and exit formalities and present a crew list.
Of course, if you’re in the wrong, they’ll make things difficult for you and demand things that wouldn’t be required if you’re following the rules.
It’s not the same atmosphere if you show up on time and voluntarily, versus being escorted by the customs patrol boat… The police and customs seem to have a certain nostalgia for the communist era and its closed borders.