You're not going to be able to access the drive through sata on computer 2 while it's still plugged into computer 1.
It's not clear what you're trying to accomplish, but it seems like burning the image to an extra laptop hard drive using computer 1 then plugging that hard drive into computer 2 would be the quickest easiest cheapest way to do this. If computer 2 is Windows/NTFS and the image is NTFS, you may have to take ownership of the (new) drive on computer 2 and change permissions on the disk to access it. If that doesn't help in your case, I think you need to give a little more information.
By "disk image" do you mean an .iso stored on the hdd in computer 1, or the image of the system hdd in computer 1?
What kind of image is it? NTFS/Windows? Something else?
Is computer 1 functional?
Is computer 1 on while you're trying to do this? If yes, can you set up a network share on computer 1, then mount the image on computer 2 with no need for the sata connection?
Are the computers Windows or Linux or something else?