  The $\beta$ Pictoris system is the closest known stellar system with directly
detected gas giant planets, an edge-on circumstellar disc, and evidence of
falling sublimating bodies and transiting exocomets. The inner planet, $\beta$
Pictoris c, has also been indirectly detected with radial velocity (RV)
measurements. The star is a known $\delta$ Scuti pulsator, and the long-term
stability of these pulsations opens up the possibility of indirectly detecting
the gas giant planets through time delays of the pulsations due to a varying
light travel time. We search for phase shifts in the $\delta$ Scuti pulsations
consistent with the known planets $\beta$ Pictoris b and c and carry out an
analysis of the stellar pulsations of $\beta$ Pictoris over a multi-year
timescale. We used photometric data collected by the BRITE-Constellation,
bRing, ASTEP, and TESS to derive a list of the strongest and most significant
$\delta$ Scuti pulsations. We carried out an analysis with the open-source
python package maelstrom to study the stability of the pulsation modes of
$\beta$ Pictoris in order to determine the long-term trends in the observed
pulsations. We did not detect the expected signal for $\beta$ Pictoris b or
$\beta$ Pictoris c. The expected time delay is 6 seconds for $\beta$ Pictoris c
and 24 seconds for $\beta$ Pictoris b. With simulations, we determined that the
photometric noise in all the combined data sets cannot reach the sensitivity
needed to detect the expected timing drifts. An analysis of the pulsational
modes of $\beta$ Pictoris using maelstrom showed that the modes themselves
drift on the timescale of a year, fundamentally limiting our ability to detect
exoplanets around $\beta$ Pictoris via pulsation timing.
