Can latios be caught with a dead eternal battrey on Pokemon sapphire?

Answer 1

With a dead internal battery (which is what I suppose you meant, and not 'eternal'), I don't think you'd be able to save the game at all. The internal battery keeps track of time based events, such as the day of the week, and once it has run dry, time based events will no longer occur. Berries will not grow, and, in the case of Pokemon Ruby, the Roaming Island won't appear. If the battery has died entirely, the game will cease to save.

Assuming your internal battery has run dry and has not, in fact, bitten the dust, no, you still cannot capture Latios. It has nothing to do with your battery, however. According to Bulbapedia, http://bulbapedia.bulbagarden.net/wiki/Southern_Island, you'd need to either mix records with someone who has used an E-Reader and scanned the physical Eon Ticket into the game, or actually scan the card yourself. E-Readers haven't been available for years. And finding the actual ticket would probably be more trouble than it's worth, if you could find the E-Reader and the ticket.