*Given that there's been no other answer to this question, I'll leave some indications from my earlier (failed) attempt to answer it.*

I think the answer to the question as stated is yes. If you include absolutely convergent integral of arbitrary products of algebraic, exponential and logarithmic functions, then the Meissel-Mertens constant is a period in your sense, and it shouldn't be very hard to prove.

In fact, Zagier and Kontsevich mention something very close at the end of their seminal paper on the topic:

- M. Kontsevich & D. Zagier, Periods (2001)

"There have been some recent indications that one can extend the
exponential motivic Galois group still further, adding as new period
the Euler constant $\gamma$, which is, incidentally, the constant term
of $\zeta(s)$ at $s=1$. **Then all classical constants are periods in an
appropiate sense."**

To see this, take the classic expression:

$$B_1=\gamma+\sum_p\biggl[\ln\biggl(1-\frac{1}{p}\biggl)+\frac{1}{p}\biggl]$$

As you mention, $\gamma$ has an integral representation. If you can prove that the sum in the previous expression is also a period, you are obviously done.

Now, I suspect that is a relatively easy exercise of **Chen integration**. This is the standard method used to prove that the Riemann zeta function (and more generally the multiple zeta function) at the integers is a period. I'm not familiar with the technique, but perhaps someone else not necessarily familiar with periods can complete the argument.