You can do this in LR! But you will have to export and create new files. But when you say
write my adjustments to the source files
you need to understand how LR deals with these edits.
IF the data is raw, you will
not be writing this to the source files, it's raw and Adobe treats them as read only. But if and when you wish to have an actual RGB document in JPEG, TIFF, etc, you'll export (render) that set of instructions and with the raw, you'll
create 13K of new images with the edit. No reason to even think about Photoshop (I can't even imagine how long it would take just to open 13K images).
IF the images are rendered (TIFF, JPEG etc), again, only the instructions are created, the source file can have those instructions embedded but again, you need LR to render a
new iteration with the edit. And again, way faster to do this in LR than even think about Photoshop. I agree with Eyeball's recommendation for doing this in batches.