As a general rule in developing and testing numerical software, I try to eliminate as many parameters as possible. I'm sure it has occurred to you that the distance $d$ and "floor" $z$ are scalable for specified values of $a$ and $b$.
In particular the maximum of the reference "hat" function:
$$ f(x) = a \cdot \left(1 - \frac{|x|}{a/b} \right) \;\; \text{ on } \;\;(-a/b,+a/b) $$
occurs at $x=0$, where $f(0) = a$. So we can factor $a$ out of each hat function, and modify the floor $z$ to be instead $z/a$. Effectively we will take $a=1$.
Similarly the horizontal scale of distances is relative to the size of the interval $(-a/b,+a/b)$, so that if you also set $b=1$ we can treat $d$ as a multiple of the interval half-length $a/b = 1$. Knowing the solution $d$ for given $z$ with $a=b=1$ allows us to give the solutions for any $a,b,z$ with a simple scaling.
To make all this concrete, define the special case of the original problem where:
$$ F(x) = 1 - |x| \;\; \text{ on } \;\; (-1,+1) $$
and otherwise $F(x) = 0$. We reduce the original problem to the special case by taking $Z = z/a$, and then asking for the maximum distance $D$ such that:
$$ F_{\text{total}}(x) \equiv \sum_{k\in \mathbb{Z}} F(x - kD) \ge Z $$
Then an original problem with arbitrary $a,b,z \gt 0$ would have a solution $d = (a/b)D$.
To solve the special case, start with $D=2$. The sum of our functions $F_{\text{total}}(x)=\sum_k F(x-kD)$ is a simple sawtooth wave with a peak (maximum) value $1$ at every even integer and a floor (minimum) value zero at every odd integer, i.e. periodic of length $D=2$. Thus $D=2$ would be our maximum only if $Z \le 0$, which is probably not interesting. We therefore consider how to reduce the peak-to-peak distance $D$ until everywhere $F_{\text{total}} \ge Z$.
If we cut $D$ by half, so $D=1$, the effect is to make $F_{\text{total}}$ a constant function, $F_{\text{total}}(x)=1$.
Repeatedly "halving" $D$ exactly, e.g. $D = 2^{-n}$, will again produce constant functions (something that can be glimpsed in your animated GIF).
For $D = 2^{-n}$, $F_{\text{total}}(x) = 2^n$.
Therefore we can quickly "bracket" the maximum value of $D$ by finding nonnegative integer $n$ such that:
$$ 2^n \le Z \lt 2^{n+1} $$
This cuts down considerably on the effort required to find the maximum distance $D$. But more can be said, so that eventually a fairly explicit expression for $D$ in terms of $Z$ can be given.
Despite the infinite number of summands appearing in the definition, $F_{\text{total}}(x)$ gets
nonzero contributions at any particular real $x$ from only finitely many summands. The regular
spacing $D$ between hat functions $F(x-kD)$ allows us to bound the number whose supports contain
a common point. If $|kD| \gt 2$, then $F(x)$ and $F(x-kD)$ have supports with empty intersection,
so a particular point is in the support of at most $\lfloor 2/D \rfloor$ hat functions. From this
"local" finiteness it follows that $F_{\text{total}}$ is continuous and piecewise linear.
The regular spacing $D$ makes $F_{\text{total}}$ periodic with period $D$. Since the reference
hat function $F(x)$ is even:
$$ F(x) = F(-x) $$
$F_{\text{total}}$ is also even and symmetric about $x = D/2$:
$$ F_{\text{total}}(x + D/2) = F_{\text{total}}(-x + D/2) $$
By its periodicity it suffices to consider the restriction of $F_{\text{total}}$ to $[0,D]$, and
its minimum there (resp. its maximum there) will be the global minimum (resp. global maximum).
Indeed because peaks of our hat functions occur only to the left or to the right of $[0,D]$, the
summands' restrictions to it are convex (concave up). It follows that $F_{\text{total}}$ is convex
on $[0,D]$. Thus by the symmetry already noted, the minimum occurs at the midpoint $x = D/2$ while
the maximum occurs at endpoints $x=0,D$.
The supports of hat functions containing the minimizing argument $x = D/2$ lie symmetrically to its
left and right. On the left $D/2$ is in the support of $F(x+kD)$ for $k \ge 0$ if and only if:
$$ \frac{D}{2} + kD \le 1 $$
$$ k \le \frac{1}{D} - \frac{1}{2} $$
It follows that to the left of $D/2$ we have $\lfloor D^{-1} + 1/2 \rfloor$ contributing summands:
$$k = 0,\ldots,\lfloor D^{-1} - 1/2 \rfloor $$
and an equal number of equal contributions to the right. Thus:
$$ \begin{align*}
F_{\text{total}}(D/2) &= 2 \sum_{k=0}^{\lfloor D^{-1} - 1/2 \rfloor} F(kD + D/2) \\
&= 2 \sum_{k=0}^{\lfloor D^{-1} - 1/2 \rfloor} (1 - (k+1/2)D)
\end{align*} $$
Gauss's trick of summing an arithmetic sequence applies, absorbing the factor $2$:
$$ \begin{align*}
F_{\text{total}}(D/2) &= \lfloor D^{-1} + 1/2 \rfloor \cdot \left[\left(1 - \frac{D}{2}\right)
+ \left(1 - \left(\lfloor D^{-1} - 1/2 \rfloor + 1/2 \right) D \right) \right] \\
&= \lfloor D^{-1} + 1/2 \rfloor \cdot [2 - \lfloor D^{-1} + 1/2 \rfloor D]
\end{align*} $$
Despite the evident jump in $\lfloor D^{-1} + 1/2 \rfloor$ when $D^{-1}$ is an integer plus one-half, the
above expression for $F_{\text{total}}(D/2)$ is continuous there and elsewhere. Confirmation of
this is left as "an exercise for the Reader".
Note that when $D^{-1} = n$, a positive integer, $\lfloor D^{-1} + 1/2 \rfloor = n$ and:
$$ F_{\text{total}}(D/2) = n \cdot [2 - 1] = n $$
On the other hand, when $D^{-1} + 1/2 = n$ is a positive integer, then $D = (n - 1/2)^{-1}$ and:
$$ F_{\text{total}}(D/2) = n \left[2 - \frac{n}{n - 1/2} \right] = n \cdot \frac{n-1}{n - 1/2} \lt n $$
So $F_{\text{total}}(D/2)$ is strictly increasing as $D$ decreases, and when $D \in [(n-1/2)^{-1},
(n+1/2)^{-1}]$, we have a linear dependence of $F_{\text{total}}(D/2)$ on $D$.
Determining the maximum $D$ for which everywhere $F_{\text{total}}(x) \ge Z$ amounts to solving for $D$:
$$ F_{\text{total}}(D/2) = Z $$
Let $n = \lfloor Z \rfloor$. Then $(n+1)^{-1} \lt D \le n^{-1}$.
It remains only to compute $F_{\text{total}}(D/2)$ at $D = (n + 1/2)^{-1}$ to learn whether $Z = F_{\text{total}}(D/2)$ is attained for $D \in ((n+1)^{-1},(n + 1/2)^{-1}]$ or $D \in ((n+1/2)^{-1}, n^{-1}]$.
The exact answer may be found, whichever subinterval is correct, by linear interpolation on $D$, because in either case we have reduced the term $\lfloor D^{-1} + 1/2 \rfloor$ to a constant.
