The main reason to ensure a friendly Paraguay would be to cover Argentina's flank, not to develop a front. Geographically speaking, the Paraguayan-Brazilian border is one of the most impassible regions on the continent. Marching an army into Brazil isn't impossible per se, but getting it to even a medium-sized town would be a challenge, much less anything strategic. This is doubly true with Piritini and the benighted Entre Rios sitting astride the least useless route.
Hmmm. The Paraguayans could always join an Argentine-led expeditionary force, and they have a good riverine navy that could be of use in Grão Pará, but both of those could pose command and logistics issues (the latter especially, given that it would be a non-trivial problem to get the Paraguayan river gunboats to the Amazon basin). So, yes, Britain's main interest in Paraguay would be to keep it from jumping Argentina while the latter is committed in Brazil, and to secure its industrial capacity for supporting the Argentine war effort. Which still requires an assurance that Argentina won't attack Paraguay either now or when the battle against Brazil is over.
Having known Gauchos personally and with my twin brother living in the Rio Grande do Sul for two years, I feel like the Piratini culture would not develop in a way that would condone further expansionism into Brazil. The war of independence would be seen as their epic climax of cultural ascendance, not the first step of many. They would be interested in defending their sovereignty if they need to, yes, but they wouldn't be after territories.
As far as loyalty to their neighbors, because of cultural similarities I feel their first ally would be Entre Rios (another "liberated" gaucho republic), followed by Uruguay, followed by Paraguay, followed by Argentina, with Brazil as the clear antagonist. I highly doubt Piratini would be comfortable with any country taking control of Entre Rios, though they might be convinced if they needed help if Brazil were to threaten Piratini.
And so another piece falls into place. Now we know why Argentina hasn't tried to reconquer Entre Rios up to now (or, alternatively, why its attempt at reconquest failed) - because Piratini would fight on Entre Rios' side. And that also means that if the British want to sacrifice Entre Rios in order to get the Argentines into their column, that could cause trouble with Piratini. This, in turn, could lead to other complications, because there are probably thousands of volunteers from Piratini fighting in the Italian army, and Britain wouldn't want to provoke them into going home. And if, as Admiral Matt says, throwing Uruguay under the bus would be a step too far, throwing Piratini there would also be.
The more I hear about this, the more I wonder if the politics are just too complicated - maybe Britain will throw up its hands and let the southern republics fight their own private war. But the prospect of Argentina opening up a second front against Brazil and relieving the pressure on Grão Pará must still be very tempting. I'm going to have to think about this a bit.
There's an even lower tech version that might be relevant here, especially on the fronts involving Russia--or Austria-Hungary, which strikes me as having essentially the same per capita technical capabilities as the Tsar's forces do. That is, Russians and Austrians can and will make proper motor vehicles--those wacky Russians might even produce something like this before the war's over, especially if it moves east onto Russian soil in winter--but they'll always be in relatively short supply; something that requires only a typical cart, some horses--and the gun of course--must surely find a niche; it certainly did in the OTL Russian Civil War and that was in a Russia advanced some two decades beyond the starting point of this war.
I hadn't known about the aerosanis - they're certainly something the Russians might build for winter campaigning. Some of the powers might also try to use them for alpine warfare, but if the Wikipedia article is anything to go by, that probably wouldn't work.
Tachankas will definitely be used on the eastern front and in the Balkans - hell, a technical could easily become one, if the dang engine breaks down.
It would be highly, highly unlikely for Portugal to come in on the side of France (or, specifically, against Great Britain). It's a maritime nation, with a long history of relations with Britain (and an official alliance), with very little to gain indeed from an entry into the war on the side of France.
Portugal entering on the side of Great Britain, maybe. But that would be a hard pill to swallow still, since they have the lucrative opportunity to trade with both sides as neutrals.
Remember, Portugal is currently engaged in fighting against what it believes to be North German-backed Africans in southern Africa. If they get too frustrated, they might declare war.
Their strategy for the war'd be a bit questionable, though. What - they won't destroy our empire with their overwhelming force because....they're too busy destroying the empires of the much stronger powers?
I'd guess cooler heads might take it as far as a shift from pro-British to pro-French neutrality but no farther. But I suppose cooler heads don't always prevail.
Certainly, nobody in Lisbon wants to get in a war with the BOGs. But the way things are going, an incident could easily happen in the field - for instance, a frustrated colonial governor might attack an "arms shipment" that turns out to be an ordinary North German supply train - and if so, the adults will have their hands full trying to keep things from spinning out of control.
There are other possibilities too - if the BOGs play their cards right, they might be able to get Portugal to come in on their side in exchange for help conquering Katanga. Of course, that would require the BOGs (which, in practical terms, means Britain) to take on another significant commitment at a time when they're already spread very thin.
I'm actually not sure how that particular situation will play out, but by the time I get there, sometime partway through year three, I should know.
In any event, the next update will be a literary one, to tide you over until I've finished figuring out Argentina.