I can’t summon during the quest ‘engaging the empire’

From my experience with fishing in FFXV the way you entice the fish is by doing short bursts of reeling in while moving the lure from left to right to get it moving. Once the fish bites make sure you get the QTE to hook the fish so you can start the reeling in process.

After you enter the reeling in phase slowly start to reel in the fish if you start to see the red and white flashing bar stop reeling in order to keep your line from snapping. If the bar tells you to move your left stick to the left or right make sure you do as such since it builds tension a lot if you fail to move your line. Also be sure to pay attention if the fish decides to jump in the water since it will prompt you for another QTE to keep the fish on the line. Repeat until you catch that fish!

And make sure you feed that cat!

George Willcox already provided the correct source to this, but didn't answer the question.

The summons become available randomly.

As explained in the Wiki:Astral, certain circumstances increase the chance for certain summons.

Titan's chance increases the more allies are KO or in critical health. Also he needs an open area to be able to appear.

Ramuh's chance increases with battle duration and does not have additional requirements.

Leviathan's chance increases when Noctis is in critical health. She needs an open area as well, but also water nearby.

Shiva's chance increases the more allies are KO or in critical health and with battle duration. She also needs space, but it is less restrictive than for Titan and Leviathan.

Bahamut is a special case and can only be summoned at a particular point in the story.