The signatories are countries and not international bodies.
The signatories to treaties form an international body with their membership. That's how all international bodies are created. The UN is international body because it was created by the countries that signed a treaty. As is the EU. And the WTO. And NATO, etc etc etc.
Can you explain what form agreeing with the UK would take? And how would this even work given that 27 other countries in the UN are bound by the ECJ?
There are courts of international law outside the EU. They aren't going to give one fig about an EU court. They care solely about the treaties that the EU and UK agreed to.
What would be the role of the WTO?
Expropriation in an involuntary divorce bill is a violation of GATT. The WTO was created to provide remediation for such actions. Should the EU unlawfully seize UK assets, the UK can petition the WTO and the WTO, if it finds for the UK, can then ask other countries that have liquid funds of the EU to turn those over to the UK in compensation. Of course, the UK would probably take more immediate steps...
Barring the WTO, the ICJ could also intervene and find against the EU (hey look, another international body)
Why do you think civil courts do not follow such a legal theory?
What makes international courts different?
International laws are vastly different than civil courts, but if you need a civil court analogy, think of the Lisbon Treaty as a contract. The contract states the UK can leave. Absent any negotiated deal, it just quits and the contract is then void. To make an even more personal analogy, consider your contract with your ISP. Should something happens that results in either you or the ISP triggering a valid exit clause, the parties are separated under the terms of the exit clause. If you exit prematurely, you may be on the hook for an additional payment or the full value. If you exit under a clause that doesn't require you to keep making payments,
your ISP can't come seize your car (legally) because you made a commitment to pay your bills earlier.
I should add that for the WTO and the ICJ, they can really only authorize action. They're not the cops. They can't force the EU to do anything. But if the EU starts unlawfully seizing other countries assets in contravention to the treaties they've formed with other countries, they'll find themselves out of favor with all of those other countries.