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A manuscript that lay unnoticed by scientists for decades has revealed that Albert Einstein once dabbled with an alternative to the Big Bang theory, proposing instead that the Universe expanded steadily and eternally.
Thanks!
Proposing modifies alternative. A parallel construction would be
... an alternative to the Big Bang theory, which alternative instead proposes that the Universe expanded steadily and eternally.
Instead is an adverb that emphasizes the contrast between the the original concept and the substitute concept. Its omission would not change the sense of the sentence but it would sacrifice a little of the surprised tone.
There is a reasonable argument that proposing could grammatically modify manuscript. However, since the alternative is the substance of the manuscript, there is no semantic difference.
The subject may also be Albert Einstein:
Albert Einstein once dabbled with an alternative to the Big Bang theory, in which he was proposing + that-clause.
But it does not matter greatly, whether you see as subject Einstein or alternative.
As to a rule either such a shortened relative clause with participle refers to the last noun or if that makes no sense you have to decide by logic.
As such participle constructions can easily be unclear one should be careful when using them and make sure that the construction is unambiguous.
instead
This word means literally "in the place of": In place of the Big Bang theory Einstein suggested another theory.
