If the dog is not suffering, and still has his happy dog face on and enjoys time with the family--it would be a little difficult to put him down.
If you put him down because of inconvenience, the that may be a little selfish. We take the animals into our lives and promise to care for them and love them--so what is important here? It is important that the dog is not suffering, still enjoying life. Maybe you can fix a nice area for him outside.
His skin may stink due to a thyroid problem and the ears can be cleaned and treated with an antibiotic cream. You are moving to a new home yet really cannot afford a visit to the vet? Probably under $50 or even less. You can get the price before going.
The vet will charge you to put him down so why not spend that same amount on having him checked out and get the advice of the vet re: putting him down.
They love us and trust us so please seek a vet's advice.