For me an engine failure in a twin means landing ASAP ie after all the checklists have been worked through and you are ready for an approach and landing. Landing above structural limits is a minor issue when compared to having subsequent issues with the one good engine you have left while waiting for the gross weight to work it's way down the the Max Landing Weight.
The only thing that I would say is check the landing distance required. You might be wanting to land on a wet or contaminated runway, or other braking systems might be unserviceable or reduced in functionality such as antiskid or spoilers.
If I remember correctly, the Swiss Air MD11 that went down in Canada did so partly because the Captain insisted on reducing the weight to below MLW rather than simply getting it on the ground straight away when they surely had the braking performance to do so.
That is an extreme example, but makes the point. You would look pretty stupid if you were down to one engine and you were holding 20 nm from the nearest airport dumping fuel or flying around with the gear down, and the good engine's oil pressure light came on.