Use a different Mac temporarily

If your goal is to be able to work with all of the images you’ve been using in Lightroom Classic on your current Mac, then simply use your external drive to bring your current catalog(s) and original images to the other Mac. You were not specific about how you use your external drive, but basically, as long as it has enough space for both your Lightroom Classic catalog folder(s) and all of the original photos, just bring that and plug it into the other Mac.

If your Creative Cloud subscription is for a single-user license, then you can easily install Lightroom Classic on the other Mac because two activations are allowed. Leave your current installation alone, then on the other Mac, download the Creative Cloud app, sign into it, and install Lightroom Classic. When you are done with the other Mac, sign out of Creative Cloud there and optionally uninstall the software.

You probably don’t want to use Lightroom, because it will requiring uploading everything to the cloud that you want to work with, the images will become completely disconnected from their local folders, and will no longer be able to use many Lightroom Classic features. That’s OK if you were planning a permanent migration to Lightroom, but if you need to stay in Lightroom Classic then don’t start using Lightroom for this.

There is one more thing to keep in mind: Settings and presets. Those are stored in the Library folder of your macOS user account, and the Library folder is hidden by default. If you want the Lightroom Classic setup on another Mac to match your current Mac, be sure to copy those files too. The locations of those files are in this Adobe help article:

Preference file and other file locations | Lightroom Classic

You can use Lightroom Classic. Please see: https://www.lightroomqueen.com/how-to-lightroom-catalog-multiple-computers/