In umbral calculus, what is the established value of $\operatorname{eval}\ln (B+1)$?

Question

In umbral calculus, what is the established value of evaluation (index-lowering operator) of the logarithm of $B+1$ where $B$ is Bernoulli umbra?

In this preprint the author argues it to be $-\gamma$, the Euler-Mascheroni constant (he uses notation $B$ for $B+1$).

On the other hand, we can represent umbral calculus as germs at infinity of functions with the following mapping:

$\underset{x\to\infty}{\operatorname{germ}} f(x)\to\int_B^{B+1}f(x)dx$

and

$g(B)\to \underset{x\to\infty}{\operatorname{germ}}D\Delta^{-1} g(x)=D\Delta^{-1} g(x)|_{x=0}+\int_0^{\infty } D^2 \Delta^{-1} g(x) \, dx$

where the regularization of the right-hand part works as index-lowering operator.

Code for Mathematica:

f[x_] := PolyGamma[x + 1]
Integrate[f[x], {x, B, B + 1}] // FullSimplify
g[B_] := Log[B + 1]
D[Sum[g[x], x], x] // FullSimplify
(D[Sum[g[x], x], x] /. x -> 0) + 
   Integrate[D[Sum[g[x], x], {x, 2}], {x, 0, Infinity}] // 
  FullSimplify // Quiet

This works well for polynomials, for instance, $B^4$ is mapped to

$\int_0^{\infty } (4 x^3-6 x^2+2 x) \, dx-\frac{1}{30}$

The regularized value of this integral is $0$, so, the regularized value of the whole expression is $-1/30$, the Bernoulli number $B_4$.

But if we take logarithm, $\ln(B+1)$ is mapped to

$\int_0^{\infty } \psi ^{(1)}(x+1) \, dx-\gamma$

The regularized value of this integral is $\gamma$, so the whole expression has regularized value of $0$. Thus, according to our representation, $\operatorname {eval}\ln(B+1)=0$.

It seems, an umbra with moments being Bernoulli numbers can be represented in different ways, and depending on definition, $\operatorname {eval}\ln(B+1)$ is either $0$ or $-\gamma$.

I want to underline that if we define it to be $0$, we lose some wonderful properties which otherwise are desirable, for instance, representation of derivative of an analytic function

$f'(x)=\operatorname{eval}\Delta f(x+B)=\operatorname{eval}\left(f(x+B+1)-f(x+B)\right)$

and connection between logarithms and trigonometric functions:

$\operatorname{eval} \frac1\pi\ln \left(\frac{B+1/2 +\frac{x}{\pi }}{B+1/2 -\frac{x}{\pi }}\right)=\tan x$

$\operatorname{eval}\frac1{\pi }\ln \left(\frac{B+1-\frac{x}{\pi }}{B+\frac{x}{\pi }}\right)=\cot x$

The question: is the value of $\operatorname {eval}\ln(B+1)$ established in literature on umbral calculus?