Sorry for being a few weeks late to this, but hopefully I can shed some light on how to think about it.
The ISRO launch of these satellites is legally problematic in several ways.
First, and as already mentioned in the question itself as well as another answer, one key issue here is Article VI of the Outer Space Treaty. For purposes of this launch, a key phrase here is "The activities of non-governmental entities in outer space, including the moon and other celestial bodies, shall require authorization and continuing supervision by the appropriate State Party to the Treaty." What that means is that when a commercial company wants to do something in outer space, some government has to both (1) give them permission (authorization) to go do that thing, and a government has to (2) continue to assert some regulatory control over those activities (continuing supervision) to ensure that the commercial actor doesn't run around and violate the obligations contained in the Outer Space Treaty. With this launch, ISRO and the Indian Government have not authorized the payload activities, and they certainly have not asserted any kind of regulatory control over their on-orbit operations. Furthermore, India knew, or should have known if they asked, that NO OTHER COUNTRY was going to authorize or continually supervise these operations either, and that the United States had specifically denied permission when asked by the commercial company. So there is a very strong case that India's launch of these satellites does not satisfy the Article VI obligations.
But that's not all. India is also party to the 1974 Registration Convention, which requires "launching states" to register objects launched into outer space in a publicly available registry, and to inform the UN so that other actors can keep track of objects in space. I won't go into the definitions here, but suffice to say that there can be multiple launching states under the treaty; here, potentially India and the United States, since India was the location of the physical launch and controlled the launch vehicle, and the United States because the payload was from a U.S. company. At least one of the launching states has an obligation to register space objects. In this situation, the United States Government had specifically forbidden the company from launching, so they certainly didn't see themselves as on the hook for registering. The only government that made an affirmative decision here to launch these satellites was India, and they have not registered them anywhere. So India is in violation of its Registration Convention obligations as well.
The question of consequences and enforcement leads to a third problem. Generally, the Outer Space Treaty and the other space treaties don't have direct enforcement mechanisms. What they do have, in Article VII of the Outer Space Treaty and elaborated in the 1971 Liability Convention, are rules for liability in the event of damages caused by a space object. India is a party to the Liability Convention. The Outer Space Treaty and the Liability Convention, taken together, provide (very simplified here) for launching state liability for damages caused in orbit if there is "fault", and for absolute liability caused on the ground regardless of fault. If these satellites were to cause damage to other objects in orbit - and remember, the FCC denied them because they were so hard to track - then the launching state or states could be on the hook for huge sums. Theoretically, some might consider both the United States and India to be launching states for liability purposes, but the Liability Convention provides that in a situation with multiple launching states, "the burden of compensation for the damage shall be apportioned between the...States in accordance with the extent to which they were at fault..." These claims can go to international arbitration if they aren't resolved diplomatically. In this situation, the United States Government denied a permit and tried to prevent this launch, while the Indian Government did not seem to do any kind of safety or other due diligence checks. If there were an incident that caused damage, the "fault" - and thus responsibility to pay - seems likely to rest with the Indian Government. Despite that fact, we haven't seen any recognition of the potential legal risk by India, at least not publicly.
Edited to fix typos.