I’ve always heard it Ar-gen-teen-a or the Ar-gen-teen tango.
Until the host on Dancing with the Stars announced it as Ar-gen-tine (ryhmes with twine) Tango. Bugs the heck out of me. How in the heck could you say Argentina any way except teen-a? It’s Tina just like the woman’s name.
Spelling doesn’t always indicate pronunciation. Especially with Spanish and French.
How do you pronounce it?
I don’t know the name for it, but I do know of the Ar-gen-TYNE pronunciation of Argentine. I’ve always thought of it as an older way of saying it. Though I could also see it as a British way of saying it, with them anglicizing words more often.
I think I’ve always considered that just another way of saying “Argentinian” as in, that something comes from Argentina.
i.e. “That tango is Argentinian”
or “That tango is from the Argentine”
Neither would raise an eyebrow with me though I suspect “The Argentine” has a specific meaning that I can’t be arsed to look up
The country is Ar-juhn-TEE-nuh.
The adjective is AR-juhn-tine (to rhyme with twine).
(In British English, anyhow.)
Edit to add: An alternative adjective is “Argentinian”, but “Argentine” is more common. As a noun, “the Argentine” is also a rather old-fashioned way of referring to Argentina.
I have heard both used in Britain. I have an impeccable cite - Andy Cameron performing “Ally’s Tartan Army” . He says “the argenTEEN”. That was 1978 though, and yes, it was probably a little old-fashioned even then.
I dance Argentine tango. I use the “twine” one for the adjective. I’m from Europe and learnt most of my English in Britain.
Glad to hear someone else pronounce it Ar-gen-teen Tango. I was beginning to think I was the only one. 
I use a hard R. Arrr! like a pirate.