Usually, lui replace a "complément d'objet indirect" so yes it is replacing maire. It could be also formulated as L'écharpe de maire barrait sa poitrine without going through 'lui', but that would be vague without any other explanation about the person whom the écharpe is barring the chest.
Yes lui refers to the mayor.
Your sentence is understandable but would likely be stigmatized because it looks like you translate "Camille's chest" by "la poitrine à Camille" while "la poitrine de Camille" is expected.
You can then say:
L'écharpe de maire barrait la poitrine de Camille.
or just reword the original sentence that ...
Your apprehension of this matter is correct.
It is the subject of a subsection in LBU 14th edition (§ 672 c)). It is possible to download a pdf of this source here: https://www.pdfdrive.com/le-bon-usage-grammaire-fran%C3%A7aise-14e-%C3%A9d-d157189845.html.
Here is what can be read there. (Capital letter divisions, bold type and italics due to user LPH;
Body parts and possessives
From A Comprehensive French Grammar, 228-229,
use the indirect object pronoun to refer to the person affected when the action applies to someone else's body. The indirect object pronouns (complément objet indirect) are me, te, lui, nous, vous, leur.
Il m'a tordu le bras. He twisted my arm.
Elle lui lave les cheveux. ...
I’m tearing some off of it would theoretically be translated to:
(incorrect:) J'en en arrache.
but this sequence of two joint pronouns (pronoms conjoints) is agrammatical.
Assuming you really don't want to repeat any of the complements, the way to handle this conflict is simply to merge the pronouns in a single instance with what is called an haplology (...
The sentence is ambiguous but the second meaning ("we all want to save them") is not likely to be right so I would rule out a mistranslation.
If that were intended, we could say:
Nous tous, on veut les sauver.
To convey "we want to save them all" in everyday speech, I believe « nous voulons les sauver tous » / « on veut les sauver tous »...