I am trying to merge/dissolve these two faces, but so far any option just removes the central vertex and dissolves it into an "average" face. How do I just merge them as if I removed the intersecting edge only?
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Edit mode> Face selection> RMB + SHIFT the desired faces> F
Rough end result:
If the underlying structure is more complex than your orthographic view suggests, adding an inconspicuous loopcut or two may simplify things (so long as it won't be affected by doing so).
Hopefully this is what you were looking for.