Hi there! Thanks for reaching out. You're right that I've never tested Limited Inventory with the Grid Inventory plugin by SangHendrix, so compatibility issues might be the culprit. For reference, Limited Inventory is compatible with some VisuMZ plugins, such as VisuMZ_4_VisualItemInv—you might consider using that as an alternative to see if it meets your needs.
I think the issue is due to not having an inventory check in place. Please read the plugin instructions carefully because you need to do a conditional branch with a script call check before giving the item using the item's ID and the amount like so:
There are unique script calls for items, weapons, or armors, so definitely check out the plugin help file for more information. Also, don't forget to choose the inventory type you want to use - you have Item Weight or Slot Based as options, and both behave quite differently. Thanks for your support and I hope this helps!