I'd say something like this could really work for the first movie of a new trilogy. I think the corporate espionage angle should definitely be a part of it, and the reason for why the Dinos start becoming a problem again. Base it around the idea of Pterodactyl and Plesiosaur sightings at first, and then the sudden unexplained rampages tying into the moles within the company idea and we have a winner!
I also like your idea of keeping the plot mainly on the cargo ship; away from the mainland and the island for a lot of it. That, and the rampage idea could definitely take the trilogy away from the park and break up the formula while using some of the ideas in a new way. Your ideas really could tie together a lot of the other suggested plotlines. I particularly like the idea of Tim and Lex running the company and having a stake in the appearance of dinos on the mainland.
Maybe put a twist ending on it where at the end it's revealed that the rival company, BioSyn, had lots of carnivores on the mainland being prepped to be placed in their own parks in many countries. And the survivors Tim, Lex, Saddler, and Ian Malcolm witness another rampage already in process the second they return to the mainland. So they know something's up and BioSyn had more on the mainland.
You could even build upon the idea of the Dino rampages suddenly appearing by turning it into an outbreak theme that runs through the new trilogy. Making it much darker. Where these cloned frog-hybrid Dinos start reproducing at an excessive rate instead of getting smarter like the silly idea they had for the raptors to use guns at one point.
And by the third movie it's us who are almost extinct... and the middle or end of the trilogy can be called extinction when it's basically us vs them, and them vs themselves as they eat each other to survive and keep reproducing. Then it could really become a dark version of Cadillacs and Dinosaurs with Tim and some new survivors he finds as main characters.
If they still really want to make the raptors super smart, then hopefully they don't give them guns, however the second movie could tone down that idea and use it in a more realistic way. Making it almost like Rise of the apes. The rise of the smarter and even faster Raptors. Multiplying and dominating.
The gov'ts and armies of the world don't know what to do when many cities start having major problems similar to what was witnessed at the start of the trilogy; so they bomb the islands, but hesitate just a little too long when mass panic erupts in many cities. They try a diff solution that backfires even worse and only speeds up the transformation of Earth into Jurassic World. Really moving the story away from the parks. It could be all about the military response/inability to bomb cities leading to the loss, exploding dinosaur populations everywhere, and mass groups of raptors recognizing military locations and disrupting the ability to strategize, navigate, or exterminate the exploding populations.
Becoming an almost higher class of the dinosaurs that have now allowed similar rampages and escapes to go forth worldwide. And an even larger threat in the third movie as the most intelligent species of dinos who keep Tim, the other humans, and many of the other dinosaurs on the run.
We already know that the dinosaurs have found a way to reproduce.. despite being genetically altered to not be able to.
Quite possibly because of the frog DNA.
They could become a plague on the world.
Have you seen that episode of the Simpsons where Bart introduces a foreign species to a new environment?
Sometimes when that happens the foreign species multiplies at a hard to handle rate. In this case it's a hybrid dino-bullfrog species.
So a paleobotanist can come back into it and tell us how the environment was perfect for them back at the parks... but on the mainland the bullfrog genetics have allowed them to adapt to a new climate and new ecosystems in unexpected ways.
Multiplying across the world like the frogs Bart lets loose in that Simpsons episode. All because of Ingen's mistake in using the frog genetics (which was the secret the corporate moles within the company spent years learning, while others were raiding site b). Also because of BioSyn trying to actualize Hammond's goal of putting a park in every part of the world. Like a Jurassicland or Jurassicworld.