Well it should have been obvious by just looking at the "Unique" modifier added to the Speed Belt replacement, but I have now unfortunately confirmed that you cannot obtain a Blitzring from King Arthro if you already have one with you.
(Context: Blitzring is a tradeable item worth about 15m from an NM in Eureka Pagos, and is mostly an equivalent replacement for the former Speed belt drop)
Naturally, if you own the BIS ring inside of Eureka, you're going to be using it while inside of Eureka, however due to the fact that it was made a "Unique" item, using it inside of Eureka Pagos now excludes you from being able to get a second one as a drop, despite it being an incredibly valuable tradeable item.
Yes, you can store the ring inside of a retainer or chocobo saddlebag, however you can only access those outside of the instance. If you are already inside of Pagos when King Arthro spawns, you would have attempt to leave the instance, store it quickly in your chocobo saddlebag, hope you re-enter the same instance, and then make it back to King Arthro in time, knowing that he typically dies approximately 20 seconds after spawning.
But why is the ring unique? In the name of keeping players from having an extra +3 haste bonus in content from two expansions ago? Meanwhile there's been silent buffs to the Double Edge, Protect and Shell logos actions far greater in the player's favor than a 3% increase to attack speed.
The other option then is just never using the rare BIS ring that drops inside of Pagos, while inside of Pagos, which seems like an unnecessarily player-hostile way to prevent such a minor buff to DPS afforded by being able to equip a second Blitzring.
How can we get this unique tradeable ring decision reconsidered?
If the ring being Unique is so important could King Arthro not just drop the existing non-unique Speed Belt token item that can be traded for the unique Blitzring?