Should light infantry units have military working dogs?

You raise some valid points. In our recent wars the units at the tip of the spear have been living mostly out of combat outposts in austere environments for long stretches of time. It's crappy and stressful. Aside from the tactical benefit, having a dog might just be good for all around moral.

The cool guys run their own in house dog programs. The rest of us just have to request the asset from a supporting unit with the capability. It's not an capability that regular Infantry need enough of to make it fiscally worth while.
Then you have the actual employment of the dog. Tactical employment of a dog has special considerations. If the dog get tired it is pretty much worthless. So most Infantry units walk too far to take the dog with them. Not a problem when you have your own lift assets like a special operations raid.

So there is a lot that goes into the working dogs and maintaining their certifications, training and well being.
I only know about the Engineer Mine Dogs and their handlers as we had the course under the brigade when I was CSM. They are a great asset but come with quite a logistics trail that I'm not sure light infantry outfits are set up to accommodate.