I am definitely not a geologist, and this is probably un-provable... but, to me, it sure seemed like a plausible idea.
Your idea's first hurdle is the length of day vs the length of month. How do you keep the moon over one bulged part of Earth at all times? (it's a very different story for the moon, whose 'day' is as long as its orbit, more at [1]) Additionally, there is a solar tide at work on the oceans that is visible daily, and especially at spring and neap tides. The sun's mass drives a maximum of almost a third of the total ocean tidal bulge.
The second hurdle is viscosity (or if you like, elasticity): tides bulge the most readily-flowing parts of a body the strongest. That means the atmosphere (which is more strongly affected by daily heating <https://en.wikipedia.org/wiki/Atmospheric_tide>), and the oceans are much more strongly affected than the "asthenosphere" (the rocky parts).
In particular, tides bulge the ocean by about a metre. The solid earth tide is closer to 0.2 metres. Thus, the question is: how do you raise a tidal bulge in the rocky part of Earth and keep it from being flooded by ocean tides?
> I am definitely not a geologist
It's more gravitation vs the liquid flow of a stratified (layered) planet than geology that is are the big hurdles.
For experts, tidal Love numbers are the important things for any round stratified body: <https://en.wikipedia.org/wiki/Love_number>, which describe the bumpiness (mass multipole) raised by tidal fores on a spherical body immersed in a tidal gravitational field.
The mantle's rheology (elasticity, viscosity, rigidity), representing about 85% of Earth's total volume, is the primary driver of the Earth's Love numbers. There's more detail on that here: <https://geodesyworld.github.io/SOFTS/solid.htm#link3>.
The crust is quite low-volume by comparison. So a third hurdle is straightforwardly: if there were a persistent bulge in the mantle, why wouldn't a relatively thick part of the crust (a giant continent in your idea) not just "roll downhill", or conversely, how do you get fractions of a round supercontinent to "roll uphill" to some lunar-attraction-induced meeting point?
Finally, with present understanding of continental drift, the continents tend split apart and come together over millions of years, and there's no evidence for anything approaching circularity. The link at the top lets one step through almost a billion years of continental drift, showing this fairly clearly.
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[1] Slide 8 ("Tides (2)") of the lecture notes at <https://websites.pmc.ucsc.edu/~fnimmo/eart162_10/Week8.pdf> is pretty accessible, showing an Earth-centric diagram of tides and a moon-centric diagram of its tides, and throughout the rest of the slide deck there's lots of heavy stuff for people who like to grapple with mathematics.
[0] https://physics.stackexchange.com/questions/31429/how-long-w...
[1] https://www.dummies.com/article/academics-the-arts/science/p...
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[1] You could probably toy with models of Earth-moon as a pair of Jacobi ellipsoids or piriforms (pear-shaped, thin ends inwards) but I don't see that working without a much smaller mass ratio and higher spins. Piriform bodies (at least of homogenous self-gravitating fluid, which is a good representation of the mantle) are generally unstable. Maybe that's good if you can find a path that relaxes back to a Maclaurin (oblate) spheroid for the Earth mass that doesn't also relax the (whole of the) "bump", and relaxes the moon to its weak Jacobi (scalene) spheroid.
Really speculating substantially away from what I know: maybe the "synestia" flavour(s) of the giant impact hypothes(e)s for the origin of the moon might be a path to some test simulation codes: coalesce an ellipsoidal (or as I said, piriform or even oviform) moon first and have that drive some aspects of Earth's planetary differentiation (which happens later in that (family of) model(s): <https://en.wikipedia.org/wiki/Synestia>). In particular, the driving should be away from homogeneity in an attempt to escape eventual hydrostatic equilibrium for the Earth-mass which otherwise leaves you stuck with encoding surface features on the (very) thin crust and then dealing with the Mauna Kea problem above. I don't know how you could approach this idea with realistic chemistry though, which I think melts & dissolves this line of thinking.
ETA: Really wild speculation: with unrealistic chemistry, freeze out a long-term solid hourglass structure with the neck at the Earth's centre of mass, piling lots of rocks on the ends terminating just under the surface (but above the mantle) at the poles, and then have one pole always point to the moon Mass. Doesn't at all fit lots of lines of evidence in very old surface rocks, though. Also very hard to wash out tides raised by the sun.