The Oxford Advanced Learner's Dictionary defines come off worse as the following:
to lose a fight, competition, etc. or suffer more compared with others
I am wondering whether "compared with" can properly be used with "more" in standardized tests such as the SAT (and therefore in standard written English).
The other answers have addressed the matter of correctness and parsing: the phrase is 'suffer more', and that's compared with 'others'. This answer addresses the issue of why the phrasing works, given that the phrasing 'suffer more than others' is more familiar to you.
Choice of phrase often depends on context. Here, the context is a dictionary entry for the phrase "come off worse". The word worse is a very important component of this phrase. Since it is fundamentally a comparison, the example highlights the comparative aspect ("compared with").
The phrase "suffer more than others" highlights the suffering of the unnamed person. Changing than to compared with shifts the stress to the comparison.
In English, stress is sometimes expressed by using 'less common' or 'more unusual' phrasing - the phrasing is said to be marked.
In linguistics and social sciences, markedness is the state of standing out as unusual or divergent in comparison to a more common or regular form. In a marked–unmarked relation, one term of an opposition is the broader, dominant one. The dominant default or minimum-effort form is known as unmarked; the other, secondary one is marked. In other words, markedness involves the characterization of a "normal" linguistic unit against one or more of its possible "irregular" forms. - wikipedia
It's arguable that 'compared with' is the marked form, compared with 'than' in that context. Alternatively, one might argue that the stress was created by using the longer phrase, or simply that the comparison was made explicit by using the word "compared". Whichever the case, the focus is on the comparison.
Finally, a note about the parsing. Having read the quote several times, I think the natural parsing is:
In particular, "compared with others" only applies to "suffer more", not to losing the fight and so on.
You're parsing the sentence incorrectly. It's not more compared with, as a phrase, but fight, competition, etc. or suffer more that fit together as a phrase. It is a list, clarified below.
That said, etc. and suffer more would seem redundant... but who am I to argue with OALD? I'd surely come off worse.
In general, comparisons use "more than". Alice suffers more than Bob. You might hear someone say "Alice suffers more compared with Bob", but it's not standard.
The definition you quoted is different because it doesn't mean "suffer more than others" but rather, "suffer more, in the context of being compared with others". That's why they used the modifier "compared with others" to modify the whole sentence.
What the lexicographer is trying to say is this:
To come off worse is to emerge from something in a worse state as compared with others, that is, not as compared to one's own former state. It means to fare worse than another who has undergone the same thing.
Candidate K. came off worse in the debate.
