It's quite correct that punctuation only really became at thing after the printing-press. So it's be perfectly natural to go without it in many fictional settings. Even so, even though it was anachronistic, I worked on this question for my own last conscript because I enjoyed it.
I read what I could to find all the "functions" performed by every punctuation mark I could find. Then I picked which functions I wanted for my language.
I added one: a sentence-opener mark, in place of capital letters. For the river-inspired conscript in question, this mountain mark's slight angling helps orient the reader to the "river's" direction, too.
I remixed some: splitting the functions of semi-colon; combining em-dash, colon, and quotation under the function "hard parenthetical"; ending sentences/rivers with the "hard parenthetical" "cataratct" is an exclamation.
I left one out: questions are indicated by a clitic already, so there's no question mark.
In my opinion, my punctuation system relexes even less because many punctuation functions are done by how you write the thoughts with respect to each other, like rivers' tributaries. Information encoded by the angles of clauses isn't exactly punctuation, but perhaps it will suggest more possibilities for using text without relexing.
I could have relexed even less, I feel, because there are more nuances and "functions" languages could express by punctuation, though I don't know a language that does. Cantonese has grammar particles doing interesting tasks. English uses prosody for many expressions, and we write this with italics and such. I am sure other languages have features like these. Grammatical nuances like these could be done by punctuation in writing. It depends on what the bare sounds themselves leave ambiguous.
Setting other languages aside, have you found some ambiguous sentences in your conlang? Every language's rules leaves ambiguity somewhere; you just choose where. Perhaps your language has ambiguity in novel places, and so in the less expressive writing media would need a novel mark.
You could also have not-exactly-punctuation symbols. Consider where you might have a culturally important mark like $, or a short-hand mark like &. Culturally important information, short-hand: other common marks are surely potential hiding categories of punctuation job in front of your face :) In my conlang, the first time you mention the name of a god in a conversation or chapter or whatever, you have to use a respectful prefix reserved for this purpose. That "piety function" could have been done by a punctuation instead of morphologically. Some cultural quirk like that in your setting could have a punctuation mark in print :)
Hope one of these ideas helps!