Luke, I have to say your search for a buddy should stay as just a search until your wife passes. She does not need another animal to care for while you are at work. Especially a puppy. And leaving a puppy out back with your other dogs while you are gone is not a good solution. The pup could easily be harmed, or could harm your small dog with nothing more than playing.
Your wife wants a lap dog, why not focus on what your wife wants while she’s still with you.
I’m in a world of hurt at my own hands right now. We had two dogs. Then we purchased a pup from a breeder. Then found out I would need surgery now, instead of two years from now. Even knowing I was heading into surgery soon, I said yes to an emergency foster. 10 month old Belgian Mal. Which DH fell in love with and promised he would do everything, all the work and training he and our older male needed. He hasn’t lived up to that agreement, and it’s been **** on me. I’m either looking at training 3 dogs while in recovery, or rehoming because it’s way too much for me to handle physically.
I 100% understand the need to focus on something positive (a new puppy in this case), than it is to deal with the emotions you must be going through, but trust me, it’s a decision you will come to regret. I certainly do, and I haven’t been given a terminal diagnosis.