Since this is a big thread on the War of the Triple Alliance, what might Paraguay's demographics have been like without that conflict being so horrible and within the modern day borders--I've been wondering? An absolute massacre/famine and severe gender imbalance does not suggest the demographics were very good for a few decades, and I've seen the status of the Guarani language in modern Paraguay attributed in part to the devastation. So would Paraguay have similar or more population assuming the same conditions in Latin America in the 20th century assuming Paraguay didn't suffer as bad in the conflict?
White peace is definitely Paraguay's best goal. Brazil and Argentina can outproduce them and more easily rebuild a fleet, not to mention the manpower needed. But that will take time, and Brazil and Argentina might just throw their hands in the air and take Lopez's peace offer. I've mentioned WWII Japan a lot in this thread, because they're so similar. Nations that fought to the bitter end (Japan easily could've have in 45-46) for goals so out of reach, even though they could bunch well above their theoretical power. That's why Paraguay must be very careful, but at the same time, Imperial Brazil in the 1860s isn't the United States in the early 1940s--it's a country that could easily settle for a white peace. You're right about Solano Lopez--he's a lot less insane than he's portrayed as in many accounts. He's one of history's gamblers--take his contemporary Robert E. Lee for another man who gambled a ton on battles and won more often then not (until he inevitably lost). But Francia himself is often considered insane, even though for his context, it's difficult to find a better guy to deal with the hand he was dealt.
As far as I know with Uruguay, anything with the Colorados will mean more rounds of the Uruguayan Civil Wars, meaning the place will take longer to stabilise. Not sure what the status of the place between Brazil and Argentina was in the 19th century--like the UK and Belgium? Obviously the Cisplatine War and such, but later on? Adding a third power seems like you'd have to rewrite the political situation of the region. It seems a flashpoint and an area of instability--Paraguay would be best to be very careful, since it's an obvious place for a casus belli from Brazil or Argentina to come from.
I'm still not convinced with the need for an ocean port, and if that would be in Paraguay's best interest. Adding Corrientes and Entre Rios means two powerful Argentine provinces are now in Paraguay. That means their governors will now be some of the most powerful individuals in the country. Look at General Urquiza--right from Entre Rios. Corrientes and Entre Rios might be good, but might also be bad, and will definitely be throwing a huge wild card into play for the future of Paraguay, all for the sake of some ocean front which could be effectively secured through enforcement of treaty rights and a strong army/riverine navy. But that definitely grabs most all of Argentina's mate-producing land, so that might have some positive effects but at the same times, allow the local governors a lot of control over what happens. But in practice, they can still maintain somewhat of a parity with Brazil assuming there isn't mate planting anywhere else it might be suitable based on how it catches on (i.e., a US planter in the Deep South experimenting with it, where it'll probably grow if my research is right, the Ottomans--mate is popular in parts of the Arab world, evidently, could it be produced there, colonial efforts in Africa in the 1880s onwards). A lot does depend on how much it catches on--I don't know how much the relation of the various plants of the Ilex family were known in the 19th century, but it appears as though yerba mate and the US South yaupon have overlapping ranges if you cared to transplant the crops there. Something individuals in the Deep South might appreciate if the country ever finds itself a taste for mate. It'll probably work better than the coffee and tea plantations in the US South ever did. And that cuts into the revenue potential of mate for Paraguay and the rest of the region--what's the point of gaining all this mate-cultivating land when Georgia, South Carolina, etc. are outcompeting you?
"Banana republic" type rule as in Central America might only happen if yerba mate catches on in the US or Europe, and the right capitalists meet up and plot something nefarious out for how to gain a market. Okay, maybe not that obviously evil, but it definitely takes scheming to control governments like United Fruit did in Honduras and such, and of course plenty of money to dish out the bribes to various politicians.
I'm not too familiar with the coffee booms either, but from what I've read, that's what gave Brazil the power to do things like buy their dreadnoughts and thus spark the South American dreadnought race.