may I please ask, if atman is "like" anything
I think that:
- 'Atman' is more commonly used in Hinduism
- Buddhism usually denies that 'Atman' exists ... the more usual Buddhist doctine is 'anatman': according to Buddhism, 'anatman' is a better description of the world
- Anatman is a word whose definition is a negative (negation) of Atman.
- 'Atman' is a word but not necessarily a reality (the mere fact that a word exists doesn't imply that what it's describing exists: see for example "unicorn").
So Atman is a word with a meaning, which other schools (not Buddhism) claimed had existence.
When some schools of Buddhism were trying to describe something (i.e. 'Buddha-nature') they re-used an existing word (i.e. 'Atman') for that description.
I think it's less confusing to view/use 'Atman' as an adjective instead of as a noun.
For example, this says,
Even though he has said that all phenomena [dharmas] are devoid of the Self, it is not that they are completely/ truly devoid of the Self. What is this Self ? Any phenomenon [dharma] that is true [satya], real [tattva], eternal [nitya], sovereign/ autonomous/ self-governing [aisvarya], and whose ground/ foundation is unchanging [asraya-aviparinama], is termed ’the Self ’ [atman].
I (perhaps incorrectly) understand that as saying that 'atman' can be used like an adjective to describe anything/whatever is "true, real, and eternal".
When you ask, "ia atman 'like' anything?" IMO that's as if you're saying, "assuming that atman is a substance or a thing, can I understand it as being 'like' some other substance or thing which I already understand or have a word for?"
It might be better to say, Atman means "true, real, and eternal" and you can use it describe that which is true, real, and eternal.
is it more like the eventual non existence, annihilation, of the empirical self
If a dog is (immediately/already) not a cat, or if the moon is not a finger (or if a chariot is not an axle and an axle is not a chariot), then I think you can say that "it is not" -- without trying to say, "a dog is like the eventual non existence, annihilation of a cat" (but maybe whether or not you call it "eventual" depends on whether you believe in "sudden enlightenment").
Perhaps that question is unanswerable, because I can't see any alternative to those possibilities
The Buddha warned about the 'thicket of views' you get into by trying to understand 'self', so I don't understand why you (IMO) persist along these lines of enquiry.
The Buddha specifically said that instead of asking "what is self? what is the Buddha? etc.?", he wanted to teach people about the four noble truths (i.e. dukkha and the way to end dukkha).
Beware that although I tried to answer this I know even less about Mahayana doctrines and terminology than I do about the Pali suttas, i.e. this is my opinion and liable to be ignorant.