With a Base Stat Total of over 1,000, Eternamax Eternatus is probably not meant to be owned by any trainer.
However, by hacking it into a raid (save edit), and setting the form to 1 (instead of the default 0), after it was caught in a raid battle,
it does not revert back to regular Eternatus.
(Even if the "send into box" and party/summary screen shows a regular Eternatus)
This can also be achieved with a ROM edit: instead of having the EncounterScenario as Eternatus_3 in pkNX, if it was changed to a scenario that allows catching, such as Regigigas_Raid_Battle or Motostoke_Gym_Challenged, it could be captured, and would retain it's form.
Here's an example of me messing with the EncounterScenario:
The Eternamax Eternatus can be brought to the Isle of Armor and have it follow you, and a somewhat small version of it will follow you, when you set it to the lead in your party.
However, it will stay on the spot, and is unable to follow you. Thanks to Anubis (Sibuna_Switch) for testing this. (source video)
Important to note: loading any save with Eternatus in Eternamax mode will cause it to revert to standard form.
Having this in your save will make it illegal. As per any item in the Unreleased/Beta PKM Gallery, use at your own risk.