arse - ignore the using canvas size to increase by 100pixels - you have mixed portrait/landscape so go with something else, Stroke seems a nasty way as you'll be losing some of your image but blah, other options to make borders in batch actions are hell to explain
So yeah - I've assumed you know how to create actions and then replay them as Batch commands - if not, oops, it's easy though.
For an example I've attached a quick version I made that works. Unzip then Actions --> Load Actions. It's missing the final Export For Web though as that takes a folder location on your PC.
You can edit the individual parts and add more (export!) if you want to use it as a starting point but it'll at least show you some of the fiddly bits I mentioned to include (reset swatches, first image resizing to set resolution, using link layers to position your copyright) in all actions destined for batch as good practice.
My main question is
why do you have 2000 images needing copyright overlay?! Even for client proofs that's a helluva lot. Choose the best & bin the rest.