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So the bond association enthalpy for ionic compounds like $ce{NaCl}$ and $ce{NaBr}$ can be easily calculated from a Born-Haber cycle. But the way a Born-Haber cycle is constructed it uses info that only really applies to mono-atomic ions like $ce{Cl^-}$. So how would one calculate the strength of an ionic bond for ionic salts that contain poly-atomic ions like $ce{NaOH}$ and $ce{K_2SO_4}$. Is there a completely different method, or can a Born-Haber cycle be adapted to use a poly-atomic ion?

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).

$$
begin{array}{ccc}
ce{&M^0(g) &+ &X^0(g) &→ &M+(g) &+ X-(g)} \
&↑small ΔH_mathrm{s} &&↑small D &&↓ & \
ce{&M(s) &+ &1/2 X2(g) &→ &MX(s)}
end{array}
$$

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:

$$U = ΔH_mathrm{f} - left(ΔH_mathrm{s} + frac{1}{2}D + IE + EAright)$$

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:

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

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

1. 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.


2. Smart, L.; Moore, E. Solid State Chemistry: An Introduction, 4th ed.; CRC Press: Boca Raton, FL, 2012. ISBN 978-1-4398-4792-3.

If you truly believe that the ionic model is a good representation of the species of interest, given a structure you can calculate it analytically, or on a computer with an appropriate piece of software. Madelung energies and Ewald sums are the magic phrases here, you might like to look at

https://en.wikipedia.org/wiki/Madelung_constant

https://en.wikipedia.org/wiki/Ewald_summation

and the nice lattice energy calculator at

https://scilearn.sydney.edu.au/fychemistry/calculators/lattice_energy.shtml

These methods can fairly straightforwardly be extended to use point multipoles, which model distortions of the point charge, but eventually you probably would want to use a full ab initio electronic structure code, many of which can address periodic substances at at least the DFT level, and some allow potentially more accurate wave function methods as well. For "ionic" substances with small unit cells such calculations are usually routine.