Hello LurkerBelow;
In response to your question, yes, the AMMs with which I am familiar (A320, A330, A340, B777) deal with Overweight Landings in an entirely separate manner from hard landings.
Necessarily, the two combine in the AMM should such circumstances obtain.
Normally, the two events (hard landing, overweight landing) occur separately and are printed / handled separately in these AMMs.
The A330/A340 QRH requires a touchdown rate of descent of 360fpm or less. I know from experience, (35T above MLW in an A340) that if the touchdown is =< the MLW, no overweight landing check is required. We were on our way in an hour. Also, maintenance was able to see the data from the ACARS.
This kind of relief would be type-specific, likely from the manufacturer in coordination with the regulator. I don't know whether "Landing Weight" works the same as MTOW which is sometimes just defined as what the airline will pay for because limits aren't always structural but I have no idea how that works in detail.