The border between Bolivia and Argentina is an extremely difficult terrain for warfare. At that time the infrastructure in the region was IIRC very poor as well, posing major logistical problems to large offensives. So I guess the front can stabilize as the defending side can hold whatever the other side can throw there. The country that's smart enough to take a defensive stance, or can't mobilize their forces in time will probably hold the line while the other wastes the lives of thousands of young people. Since Bolivia is declaring war, I guess they'll take a heavy toll in the opening months. Whether the Argentinean vice president, Marcos Paz (Mitre was leading the troops fighting in the Paraguayan front) sees that or amasses whatever forces available for a counteroffensive is an open question. If he orders an offensive I think we might see a replay of the Independence War offensives and counteroffensives in the region with better armament: A useless waste of life and resources in with neither side can achieve a decisive victory.
The key player is Chile. Depending on the timing the War with Spain might be already over and Chile has a navy witch Argentina can't match. Brazil can but, given the distances involved, the issue is the reaction time. Furthermore in OTL a large rebellion occurred in Mendoza in late 1866. Also, since it would also be Chile the one declaring war, it's safe to assume that they would be ready to cross the Andes shortly after that - and the troops in the border are those rebelling. I see the Chilenean Navy seizing Tierra del Fuego to set up a naval base there ASAP, with the Imperial Navy engaging the Chilenean Navy some time after that. No idea of who can be victorious there. The other point is how Chile would act regarding Argentinean officers and citizens in rebellion or bound to rebel and vice versa. I do think that a Chilenean invasion would make loyal and rebel forces in Argentina sort of leave aside their differences against a common foe. And I mean "sort of" because it's a good opportunity for Mitre/Marcos Paz to get rid of them by simply using them as cannon fodder. On the other hand Chile can try to take advantage of the political climate in Argentina to foster rebellions and restart the Argentinean civil war.
Spain might also add their weight to the war, offering assistance to the (original) Triple Alliance, since they've finished fighting a war with Chile and Peru. For a lot of Argentineans, including those in the establishment, an alliance with Brazil and Spain would be a really uneasy exercise of realpolitk. Especially if Brazilian and/or Spanish soldiers are deployed in large numbers in Argentinean soil to counter the Chilenean invasion from the West.
I think that the situation has a small chance to put Argentina in the peace negotiation table. But I'm not sure if it would be enough without large rebellions in the country and those rebellions might precisely not happen if the rebels think they would be playing for a foreign country. Even if Chile defeats the Brazilian Navy and succeeds in their offensive in the Argentinean center west it doesn't yet have the political impact to convince hardliners. Plus, with Brazil as an ally, Argentina can field the necesary amount of troops to contain it. And if Chile gets as far as Córdoba, why would they sit to negotiate? And if they don't the war can continue for years. I guess the only way to end it fast would be a phyric Chilenean victory in Mendoza/San Juan and a phyric Brazilian naval victory near Tierra del Fuego. That way the Argentinean politics might feel scared enough to negotiate while the Chileneans might not want to loose Tierra del Fuego and might feel that they are vulnerable to a counterattack. If such a peace negotiation includes Paraguay, even if Brazil doesn't sign it - and the Emperor wouldn't feel any need to do so - the war in Paraguay also is forced to end: neither Paraguayans nor Brazilians would be able to fight each other much without being able to get across Argentina