Calculating the strength of an ionic bond that contains poly-atomic ions
With enough effort, Born–Haber cycle can be extended to polyatomic ionic solids, however it's practically never done in practice due to the lack of experimental data or because it's impossible to obtain any.

From [1, p. 117–118] (emphasis mine):


  Lattice energies cannot be measured experimentally since they represent hypothetical processes:
  
  $$\ce{M^n+(g) + X^n-(g) → MX(s)}$$
  
  However, the following reaction sequence, relating the heat of formation, $ΔH_\mathrm{f}$ of a crystal $[\ce{M(s) + 1/2 X2(g) → MX(s)}]$ to $U[\ce{M+(g) → MX(s)}]$ is thermochemically equivalent (and $ΔH_\mathrm{f}$ can be measured).
  
  $$
\begin{array}{ccc}
\ce{&amp;M^0(g) &amp;+ &amp;X^0(g) &amp;→ &amp;M+(g) &amp;+ X-(g)} \\
&amp;↑\small ΔH_\mathrm{s} &amp;&amp;↑\small D &amp;&amp;↓ &amp; \\
\ce{&amp;M(s) &amp;+ &amp;1/2 X2(g) &amp;→ &amp;MX(s)}
\end{array}
$$
  
  In this diagram, $ΔH_\mathrm{s}^\circ$ gives the enthalpy of sublimation of the metal $[\ce{M(s) → M^0(g)}],$ $D$ gives the dissociation energy, or bond energy of the diatomic gas $[\ce{1/2 X2(g) → X^0(g)}],$ $IE$ gives the ionization energy of the gaseous metal $[\ce{M^0(g) → M+(g)}],$ and $EA$ gives
  the electron afﬁnity for the formation of the gaseous anion $[\ce{X^0(g) → X-(g)}].$
  The lattice energy is obtained through the relation:
  
  $$U = ΔH_\mathrm{f} - \left(ΔH_\mathrm{s} + \frac{1}{2}D + IE + EA\right)$$
  
  One difﬁculty with using a Born–Haber cycle to ﬁnd values for $U$ is that heats of formation data are often unavailable.
  Perhaps the greatest limitation, however, is that electron afﬁnities for multiply-charged anions (e.g. $\ce{O^2-}$) or polyanions (e.g. $\ce{SiO4^4-}$) cannot be experimentally obtained.
  Such anions simply do not exist as gaseous species.
  No atom has a positive second electron affinity; energy must be added to a negatively charged gaseous species in order for it to accommodate additional electrons.
  In some cases, thermochemical estimates for second and third electron afﬁnities are available from ab initio calculations.
  Even so, if there are large covalent forces in the crystal, poor agreement between the values of $U$ obtained from a Born–Haber cycle and Madelung calculations can be expected.


Already for some simple species such as oxoanion it is impossible to determine their electron affinity $EA$ in gaseous phase:

$$
\begin{align}
\ce{O^0 (g) + e- &amp;→ O^- (g)}\label{rxn:R1}\tag{R1} \\
\ce{O- (g) + e- &amp;→ O^2- (g)}\label{rxn:R2}\tag{R2} \\
\hline
\ce{O^0 (g) + 2 e- &amp;→ O^2- (g)}\label{rxn:R3}\tag{R3}
\end{align}
$$

Enthalpy of reaction \eqref{rxn:R2} cannot be determined experimentally and can only be derived from the lattice energy $U$, which causes Catch 22 situation[2, p. 61].

References


Lalena, J. N.; Cleary, D. A. Principles of Inorganic Materials Design, 2nd ed.; John Wiley: Hoboken, N.J, 2010. ISBN 978-0-470-40403-4.  
Smart, L.; Moore, E. Solid State Chemistry: An Introduction, 4th ed.; CRC Press: Boca Raton, FL, 2012. ISBN 978-1-4398-4792-3.