We apply two methods to estimate the 21~cm bispectrum from data taken within
the Epoch of Reionisation (EoR) project of the Murchison Widefield Array (MWA).
Using data acquired with the Phase II compact array allows a direct bispectrum
estimate to be undertaken on the multiple redundantly-spaced triangles of
antenna tiles, as well as an estimate based on data gridded to the $uv$-plane.
The direct and gridded bispectrum estimators are applied to 21 hours of
high-band (167--197~MHz; $z$=6.2--7.5) data from the 2016 and 2017 observing
seasons. Analytic predictions for the bispectrum bias and variance for point
source foregrounds are derived. We compare the output of these approaches, the
foreground contribution to the signal, and future prospects for measuring the
bispectra with redundant and non-redundant arrays. We find that some triangle
configurations yield bispectrum estimates that are consistent with the expected
noise level after 10 hours, while equilateral configurations are strongly
foreground-dominated. Careful choice of triangle configurations may be made to
reduce foreground bias that hinders power spectrum estimators, and the 21~cm
bispectrum may be accessible in less time than the 21~cm power spectrum for
some wave modes, with detections in hundreds of hours.