Maximum tidal heights vary widely across the globe, from 16 m in the Bay of Fundy to mere centimeters elsewhere. These variations are due to coastline and shoreline differences. This makes it difficult to determine a global average ocean tidal maximum height. Let's assume the Earth is a landless planet covered throughout by an average one km of ocean. How can we calculate the maximum ocean tidal height from a new moon or full moon syzygy, assuming semi-major Moon & Sun distances?
I think you would have to get into the fluid dynamics. A uniform ocean approximation would help a lot. With any luck rather than solving for a fully 3D time dependent flow, an assumption that the result is an expansion of a few well choosen spherical harmonics, with the same periodicity as the tidal frequency might yield a closed form solution (if you are lucky).
The variations are not just "due to coastline and shoreline differences" although there is the famous example of the Bay of Fundy. They are also due to the fact that the oceans are constrained by large land masses, so that they are essentially shallow bowls of water being "rocked back and forth" by the Moon and Sun. There are several locations in the middle of the Pacific that have very low tidal excursions because the global scale 2D standing wave has a node in that region. These notions are referenced fairly well in the Wikipedia article on Amphidromic Points
The mechanism of tides is convincingly explained as a "fake force" generated by integrated "squeezing" across planetary sized water masses in this PBS Science Youtube video.