Hei Deliciae! I prefer "jenten" as my mothertongue was "Bergensk", 80-years ago! And I learned nynorsk in the "Folkeskulen", so I have some problems mixing bokmaal, nynorsk and dialect 10 miles south of Bergen, living abroad since 60 years. Tusen takk for ditt arbeid, beste hilsener til gamlelandet!
"Jenta" in both, but by a bigger margin in Trondheim.
However, most people would say "en jente" i Oslo, and "ei jente" in Trondheim for the indefinite singular. Consistency is not our forte. ;)
Keep in mind that "jenta" is a particularly strong feminine word, most feminine words would still follow the masculine declination pattern for their definite form in the Eastern dialects.
In Trondheim, the "-a" ending is used across the board, unless you speak "fintrønder" which is a sociolect reserved for rich old ladies.
"Pike" has strong Danish connotations, and the strictly Norwegian indefinite article "ei" doesn't go too well with it. I would avoid both "pike" and especially "ei pike".
To me, "pike" is more reminiscent of old movies and books where people talk about little girls as "young ladies".
Well, this is just my opinion. Others will have different opinions.
I only ever studied Bokmål, but I can usually read Nynorsk. Couldn’t write it, though.
So, if you plan to predominantly read and write Nynorsk, maybe it makes sense to study Nynorsk as your first written language, so that the “usually read it” becomes an “always read it”, and the “can’t write” becomes a “can write”.
That said, I don’t think I’ve ever been “confused” because of my Bokmål background. And, equally, when it comes to talking, no person who writes Nynorsk is going to be confused by my very-close-to-Bokmål spoken Norwegian. I mean, Norwegian is still Norwegian; it’s still recognisably the same language.