Indeed the seal of insurance for the Chocobo is now 200 seals (was 2500 in ARR), you can instantly obtain 250 seals by doing the first enemy on your new Grandcompany Hunting log, and then gather around 3000 more seals by completing the rest of that log.
I wouldn't bother with FATEs myself though that's an option, since the Hunting log for your Company gives enough, and once you get to the next tier of that hunting log you should be close to unlocking Expert Delivery which makes seal gathering extremely quick and easy through Dungeon gear exchange. And as mentioned above you can do both Guildhest Roulette and Leveling Roulette for bonus seal rewards, and if you have tank unlocked to level 16+ you can get the bonus seal reward every time on leveling roulette by repeating that duty over and over until you reach the current seal cap.