The reason to add a second dog to a home really should never be "for the other dog to have company." Two dogs do love to be together and play together, but if your dog has problems with you and your wife being away from the house, those problems will persist once a second dog is added and may transfer to the second dog as well. Sorry to say that
but it's true unfortunately.
There are a lot of things one can do to make a dog more comfortable while one is away- and dogs really do need to learn this skill- it's not something that comes naturally to them. I use kongs with frozen unflavored yougart inside them and a kennel for my dog that is prone to being upset by my leaving. He could care less about me leaving once he is distracted in his kennel with the stuffed kong. Once he realizes I'm gone (which is about 20 minutes later and we've used video to verify) he no longer is upset, curls up in a ball and goes to sleep until i return.
There are many more things one can do, but basically the easiest is to not make it a big deal when you leave and return. My dogs don't get a warning that we are leaving, we just leave and when we return, I wait a bit (put down the mail, make sure I have my coat off, etc and they settle down) before they get attention. That makes coming and going less emotional and anxiety is reduced.
Hope that helps a bit.
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Chelle and the rest of the crew including, but not limited to Kita and Taiko (the shiba inu wrestle maniacs), Adi (reserved and dignified tabby cat), and all 28 reptiles