I wonder how the nationalisms would develop in the Carlist zone. While the PNV and the Lliga were off-shots of the Carlist remnants, the Irmandades da Fala gathered majoritarily a liberal/federalist tradition from the Galician provincialism (although it had a traditionalist element as well, coming from Alfredo Brañas).
Nowadays the Traditionalists defend (beside the political ultra-catholicism), some sort of republican federal Kingdom as the political organization of Spain. Having had time to evolve on their own during the XIX century, and then with the apparition of the nationalisms in a good chunk ofl of their territory, i wonder if they would had arrived at a system of federal kingdom (or personal union through the king).
Edit: Now, if we go to the I Republic, 1973, where the cantonalism exacerbated the 3rd Carlist war...
I understand that during the federal period of the I Republic, while in the southern Spain the cantonalism exploded, in the northern regions with a more developed regionalist movement, they were starting to write their federal constitutions.
So, IOTL, in less than a year, the I Republic collapses completely, and shortly after, the army pronounces the restauration of the Borbon monarchy.
I guess that the best idea to make a Carlist Spain as an independent country could be to have a bit stronger and stabler I Republic so it doesn't collapse nearly as fast. Make it last a few years, and be able to slowly regain control of the cantonalists, but giving more free reign to the Carlists in their areas.
A stabler I Republic will also remain federalist. So what we would be looking for is some kind of arrangement at the end of the III Carlist war, where the federal states of Galicia, Navarre, Cantabria, Asturias, Catalonia, Aragon, Valencia, and Basque country adopt a traditionalist constitution for themselves (Charles VII was willing to be a constitutional king), and obtain from Madrid the right to federate among themselves under a commong king. All of those states would belong to Spain only nominally, having their own king instead of the president of the Republic to assume the executive power.
After a while, and because of the further discredit the republican govt. in Madrid has fallen to, thanks to this arrangement, the Republic falls, and the Borbon monarchy is restaured on the head of Alphonse XII. But by this time, the Carlist Spain is too firmly entrenched to be removed easily, without a very bloody war. After the fall of the republic, both Alphonse XII and Charles VII consider themselves kings of all Spain, but in fact, they have no wish to enforce their claims: the country is expended after the the Carlist war, the Cantonalist war, the constant changes of government...