Fitting multiple pKa's
Given your hypothesis that the three $\mathrm{p}K_\mathrm{a}$'s are dependent on one another and that experimentally you do not seem to be able to identify which histidines are protonated and which are not at any given time, it would seem that the simplest model to use is that of a triprotic acid such as citric acid. The triple HH equation model you used is only appropriate for three completely independent $\mathrm{p}K_\mathrm{a}$'s (such that the $\mathrm{p}K_\mathrm{a}$ at site 1 is unchanged by the protonation state of site 2). 

In the dependent $\mathrm{p}K_\mathrm{a}$ model, we have four species (vs the eight in your model): $\ce{H3P},$ $\ce{H2P},$ $\ce{HP}$ and $\ce{P},$ where "$\ce{P}$" is the protein. There are three "apparent $\mathrm{p}K_\mathrm{a}$'s" none of which represents a true $\mathrm{p}K_\mathrm{a}$ of a specific site. They instead correlate to the combination of possible reactions that give a change in the overall protonation state. For example, the first $\mathrm{p}K_\mathrm{a}$ represents the change in state from $\ce{H3P}$ to $\ce{H2P},$ regardless of which of the three sites loses a proton. 

We can then write four equations starting with mass balance:

$$
\begin{align}
[\text{total protein}] &amp;= [\ce{H3P}] + [\ce{H2P}] + [\ce{HP}] + [\ce{P}]\tag{1}\\
K_\mathrm{a1} &amp;= \frac{[\ce{H}][\ce{H2P}]}{[\ce{H3P}]}\tag{2}\\
K_\mathrm{a2} &amp;= \frac{[\ce{H}][\ce{HP}]}{[\ce{H2P}]}\tag{3}\\
K_\mathrm{a3} &amp;= \frac{[\ce{H}][\ce{P}]}{[\ce{HP}]}\tag{4}
\end{align}
$$

We can also define the fraction of sites unprotonated as

$$\phi = \frac{3[\ce{P}]+2[\ce{HP}]+[\ce{H2P}]}{3[\text{total protein}]}\tag{5}$$

It's relatively straightforward to use the four above equations to get expressions for each of the species concentrations in terms of $[\ce{H}]$ and $[\text{total protein}]$ only. Substituting those expressions into the bottom equation will give you an equation to which to fit your data. 

If you post your data in tabular form, others can play around with different fitting approaches and give feedback.