Cletus' suggestion is exactly what I do when I want to merge photos taken on the trip with my main archive, which was left back at home.
But if you want to be able to take all your photos so you can work on anything, and all the photos on your desktop computer fit on the free space of the portable 4TB hard drive, you could just copy all the photos to it. The best way to do this is to first make sure they are all already organized under a single parent folder, with all photos in subfolders of that single parent folder. If the photos are organized properly, it should be possible to copy all of them to the external hard drive in a single drag (of that one parent folder). And also copy your Lightroom Classic catalog.
Once that is done, in Lightroom Classic on the laptop you would open the catalog, then in the Folders panel right-click* the single parent folder, choose Update Folder Location, and select the parent folder on the external hard drive. That should reconnect the Lightroom Classic catalog to the images on the external hard drive, and you are ready to work off the external hard drive.
When you get home, if you need to switch away from the laptop and back to your main Mac such as a desktop, and if you added images during the trip, reverse those steps. Copy the parent folder from the hard drive back to the main Mac**, copy the catalog too, open the catalog, and do an Update Location on the parent folder to reconnect it to where the images are on the main Mac. If you did not add images during the trip, you could just reconnect the catalog to the photos on the main Mac.
*If you aren’t using a mouse or trackpad set up to right-click on your Mac, Control-click instead.
**The time and trouble doing this can be greatly shortened if you use sync software that can copy only the changed files between the parent folder on external hard drive and the main Mac.