Hello nzpilot,
My understanding of the regulations is that in the dispatch fase you need to comply with the approach climb limited weight and the landing climb limited weight. If the climb gradient of the lowest minimum can than not be met and the weather is at those minimums than that airport is below minimums for you. You can either select two alternates or reduce your weight to a weight where you can comply with the climb gradient. In actual flight it becomes a different story.
At my airline a check is made before each departure by dispatch to make sure we comply with al those weight restrictions, and if any is found limiting it will be put in the operational flight plan header.
Hope this helps!