Interesting idea. I particularly like your very realistic presumption that the only way an Inca Kingdom would survive is as a vassal of the Spanish Empire. TLs that start from 1500 and have the native civilizations of the new world completely resist European domination are virtually ASB.
Given the missionary zeal of Spanish colonization as well as the drive for wealth, I suspect that the The Inca Kingdom in this TL would have to "officially" convert to Catholicism and enter into into some verey unequal relationships with the Spanish Crown regarding access for silver and gold. At minimum the Inca leadership would have to abandon the notion of Inca divinity and formally convert to Catholicism. A synchretized form of folk Catholicism similar to that in middle america might develop with some former Peruvian divinities reimagined or reassociated with Christian saints, but the whole sun god stuff would go away.
I could see also the Spanish making use of the centralized structure of the Inca and use Inca military forces as a means of expanding and controllong their control elsewhere in south america.
I suspect that Quechua would remain the primary language of the Inca Kingdom, but elites would also be fluent in Spanish.
Local elites would also see a lot of intermarriage with resident Spaniards, and gradually they would adopt Spanish styles of dress, residence, domestic architecture, etc. In fact the country might not look a whole lot different from Peru and Equador in our TL - A Hispanicized elite minority ruling over a Quechua-speaking population. The only difference would be that, ethnically, the elite would see themselves as native and descended from the pre-hispanic noble lineages.
Regarding the name of the Inca Kingdom, I see no reason the native "Tawantinsuyu" would not survive.
When or if the revolutionary/republican drive for latin american independence in the 1800's happens in this TL, I could see two possible and interesting options for the Inca Kingdom. One would be for it to be among the first to break away - as a Kingdom - and be the model for other rebellions. Alternatively, an hispanicized native nobility might see its dominance threatened by revolutionary movements and stay fierecly loyal to the Spanish crown, even to the extent of the Inca Kingdom becoming the main ally of the Spanish and using its army to put down rebellions throughout the continent. Imagine Simon Bolivar being brought to Cuzco in chains and beheaded infront of Inca Huascar VI and the Spanish Viceroy, in return for which the Inca Kingdom is granted sovereign authority to rule the Spanish colonies of south america on behalf of Spain.
Now comes some real wierd stuff. In 1900, Spain eventually severes its colonial relationship with Tawantinsuyu, which then becomes an Empire ruling all of the former Spanish lands From Colombia to tierra del fuego. In 1908, the United States fights a war with it to claim Panama for the purposes of building a canal and loses. The Incas, in cooperation with Britain, then builds the canal. in 1912, Britain and the Inca Kingdom enter into a treaty of friendship. When the Great War breaks out in 1914, the Incas enter the war as one of the Entente powers. The USA, still angry over their Panama defeat and intuitively hostile to Britain for their alliance with the Incas, starts off neutral but favorably disposed to the Central Powers. Eventually the USA enters the Great War in 1916 on the side of Germany and its allies....