A very nice question!
You just ventured into a large body of knowledge and research in computational theory around the concept of relative primitive recursiveness and hierarchy of total recursive functions such as fast growing hierarchy, which are full of interesting and mind-boggling facts.
What function class $\cal F$ do we get by adding the Ackermann function as basic function?
Well, that is one of its own kind.
Do we get all $\mu$-recursive functions?
No. What we get are far far from all $\mu$-recursive functions since all functions we get are still total recursive functions as primitive recursion preserves totality. Recall that $\mu$-recursive functions means partially recursive functions or, what is equivalent, all functions that can be computed by Turing machines.
You might ask, then, do we get all total recursive functions?
The answer is still no. What we get are far from all total recursive functions. For example, a "Ackermann modification of Ackermann function" is not included. Here is a version of that "doubly Ackermann" function.
Let us start from the usual two-argument Ackermann–Péter function.
$$A(m, n) =
\begin{cases}
n+1 & \mbox{if } m = 0 \\
A(m-1, 1) & \mbox{if } m > 0 \mbox{ and } n = 0 \\
A(m-1, A(m, n-1)) & \mbox{if } m > 0 \mbox{ and } n > 0.
\end{cases}$$
Let $f(n)=A(n,n)$ be the Ackermann–Péter function of a single-argument. Now let us define
$$D(m, n) =
\begin{cases}
f(n) & \mbox{if } m = 0 \\
D(m-1,1) & \mbox{if } m > 0 \mbox{ and } n = 0 \\
D(m-1, D(m, n-1)) & \mbox{if } m > 0 \mbox{ and } n > 0.
\end{cases}$$
Then $D(m,n)$ is a doubly Ackermann function, which cannot be obtained from composition and primitive recursion of Ackermann functions and all primitive recursive functions. This is not surprising, since a single Ackermann modification accelerates the growth of a strictly increasing function faster than arbitrarily finitely many compositions and primitive recursions.
You may continue to ask, what if we add that doubly Ackermann function? Then we are missing the "triply Ackermann functions". If you keep asking, then you will be building a growing hierarchy by Ackermann modification. Even if you have asked $\omega$ times this way, where $\omega$ is the first infinite ordinal, the growth rate of any function thus obtained is still dwarfed by most of functions in the fast growing hierarchy.
or do we get some strange sub-class?
You may call it some strange sub-class as you like. However, as you might have expected, it is not unusual at all in the field of computation theory. Here is an excerpt from paper The Ackermann functions are not optimal, but by how much? by H. Simmons, slighted edited.
5.1 DEFINITION. For each set $\Theta$ of functions, let
$PR(\Theta)$ be the clone of those functions that can be obtained from members of $\Theta$ by primitive recursion. $PR(\Theta)$ is also called a primitive recursive degree. The primitive recursive degrees measure the complexity of functions up to primitive recursive equivalence.
The "strange sub-class" of yours is just $PR(\alpha)$, where $\alpha$ is the set of primitive recursive functions and the particular versions of Ackermann function you have in mind. In that paper, "by taking a closer look at the construction of an Ackermann function we see that between any primitive recursive degree and its Ackermann modification there is a dense chain of primitive recursive degrees."
Hopefully, I have shown you a glimpse of the fascinating theory around recursiveness and growth of total recursive functions. You may want to read some of the following articles as well as raise more interesting questions like this one.
Mind blown: the fast growing hierarchy for laymen — aka enormous numbers
Ackermann function
Higher-Order Recursion, How to Make Ackermann, Knuth and Conway Look Like a Bunch of Primitives, Figuratively Speaking
