How do animals contain and dispose of bacterial endotoxins (lipopolysaccharides)?
There are too many questions embeded here to cover in a single answer. I'll summarize and answer what I interpret to be the &quot;central&quot; question.

How do animals A) locate, B) isolate, and C) contain/eliminate bacterial endotoxins?

LPS in the blood stream is rapidly bound by LPS-binding protein (LBP) and localized to the liver for detoxification and excretion (1).
A) LBP recognizes and binds the Lipid-A region of the LPS molecule (2),
B) At this point LPS is released from bacterial cell membrane and sequestered into complexes with high-density lipoprotein (HDL). This is also the stage at which free LPS can activate CD14+ monocytes, triggering an acute inflammation response (3). However, additional HDL reduces monocyte activation, suggesting that these complexes inactivate LPS by &quot;isolating&quot; it from CD14 (4).
C) Finally, the LPS-HDL complex is taken up by the liver, either into Kupffer Cells (~25%) or liver sinusoidal endothelial cells (~75%), where enzymatic inactivation of LPS occurs either by dephosphorylation of the phosphates or deacetylation of the primary acyl components of lipid A (5)
Given the hydrophobic characteristics of LPS lipid-A and the fact that it's processed in the liver, I suspect that LPS and its metabolites are eliminated from the body via excretion into the bile ducts, but I haven't found a reference in the literature to confirm that suspicion.
References:

Wassenaar and Zimmermann, Eur J Microbiol Immunol. 2018 Sep 28; 8(3): 63–69.
Tobias et al., J Biol Chem. 1989 Jun 25;264(18):10867-71.
Vesy et al., Infect Immun. 2000 May; 68(5): 2410–2417.
Flegel et al, Infect Immun. 1989 Jul; 57(7): 2237–2245.
Yao et al. J Immunol. 2016 Sep 15; 197(6): 2390–2399.