the causus belli was that Brazil and Argentina were backing one side in an attempt to over throw the gov't, while the other side (the ones in power) called on Paraguay for assistance. Brazil sent troops to aid the insurgents, which prompted Lopez to escalate the war. To get troops to Uruguay, he requested permission to cross Argentina to get at the Brazilians. Argentina said no, and Lopez took them on, too. It is not really an invasion on the part of Brazil, more meddling with the power structure.
Lopez picked a rare moment when Argentine interests aligned with Brazil's. His best chance of winning is to wait for a time when Brazil and Argentina aren't aligned. But if we have to stick with the OTL start of the war, Paraguay has to not take on both at the same time. Militarily, Uruguay is unimportant in the fight as their forces are not a factor. This means not declaring on Argentina. They could attack northward, which they did OTL, and keep on going northward. They can then hope that Argentina denies Brazilian access. The war becomes a stalemate: Brazil has no way to get to Paraguay, while Paraguay can hold on to the Matto Grasso do Sul region of Brazil. This doesn't help Uruguay any. Eventually, If Brazil wants to dislodge Paraguay, they'll need to build roads to the Northern/Eastern part of Paraguay. If this were all that easy, it would have been done OTL. So, it'll take a lot of time. Then, the war can really begin.
