Yes, you can do this in Photoshop. This will push to bottom right of any image, even if the watermark is bigger than the image.
BUT, you will need to learn to record Actions. These are the best feature of Photoshop, BY FAR, so well worth the time. It's kind of like programming your old VCR (or TiVo).
Make an Action that opens each image from a given folder (put all your images in one folder to make this easier), then imports the layer of your Watermark artwork, and then selects both the image and the watermark.
// To make this faster, preload your watermark image as an open file, and have the Action import that, so that is doesn't have to actually load and then import the watermark for each image, only duplicate it to your new image.
Now you can align right and bottom edges of both layers. You need to select both your layers (the image and the watermark) first, so that the Layer>Align functions become available. See image for where these functions are.
Use Align Right Edges and Align Bottom Edges.
...it will be roughly where you want... but might be jammed up against an edge.
So you'll have to pad your Watermark artwork sufficiently to get it to sit where you want. Or add a couple of nudges/moves of the artwork up and to the left into the Action after it's Aligned the Watermark to bottom right.