It's seems a bug is in Integrate command and Yes you are right.
Integrate[BesselK[0, (6632555543 Sqrt[xi^2])/135117312]/xi, {xi, -1, 1}, PrincipalValue -> True](* It should be Zero*)
Indefine integral:
Integrate[BesselK[0, Sqrt[x^2]]/x, x](*Cant find !*)
but:
$$\int \frac{K_0\left(\sqrt{x^2}\right)}{x} \, dx=-\frac{1}{4} G_{1,3}^{3,0}\left(\frac{x^2}{4}|
\begin{array}{c}
1 \\
0,0,0 \\
\end{array}
\right)+C$$
With Cauchy Principal Value:
Limit[((-Inactive[MeijerG][{{}, {1}}, {{0, 0, 0}, {}}, x^2/4]/4 //
Activate) /.
x -> (-e)) - ((-Inactive[MeijerG][{{}, {1}}, {{0, 0, 0}, {}}, x^2/
4]/4 // Activate) /.
x -> -1) + (((-Inactive[MeijerG][{{}, {1}}, {{0, 0, 0}, {}}, x^2/
4]/4 // Activate) /.
x -> 1) - ((-Inactive[MeijerG][{{}, {1}}, {{0, 0, 0}, {}}, x^2/
4]/4 // Activate) /. x -> e)), e -> 0, Direction -> -1]
(* 0 *)