The story is Desertion, by Clifford Simak. I read this last week, so I pretty much remember it. I'm not a fan of City, but I really liked this short story.
Setting is one of the cold planets. To explore it, people are sent out
in new bodies suited to the environment. Not one of them has returned
or reported, and no trace can be found of their fate; at least they
did not die nearby.
On Jupiter, were the gravity is high and the atmosphere is thick. Humans cannot survive as humans, but they can be changed via a machine into lopers, a native life-form. This change is safely reversible.
This partly agrees with your description.
The military leader of the base, Fowler, has sent out four or five soldiers, none of them have returned. The woman in charge of the machine that changes the people is disgusted at at the leader when he commands her to prepare the machine for two. He indicates that it will be him and Towser, his old pet dog.
In the end, the viewpoint character goes out, with a dog, and
immediately understands: his new brain is so much roomier that he
hates the idea of being squashed back into his old one. The dog can
now talk, too.
When he goes out, he finds that the climate is wonderful, because his new body is made for it. Instead of a raging storm, it is a mild mist.
And he is surprised that Towser can talk with him now, through telepathy. He also finds he is more intelligent.
This agrees with your description.
The last lines of the story are, roughly: “I'll have to go back
eventually, but not yet; they will make me be a dog again.” “And they
will make me be a man.”
At the end of the story, he doesn't want to go back. The conversation is:
"I can't go back," said Towser.
"Nor I," said Fowler.
"They would turn me back into a dog," said Towser.
"And me," said Fowler, "back into a man."
This agrees with your description.
I'd guess that this is the one by Poul Anderson, because it fits his
tragic mode, and because if he wrote it I have probably read it.
(Anderson also wrote “Call Me Joe” which shares some elements, so I
may be confused.) Can you confirm?
The story is by Clifford Simak, but he wrote in about the same time-frame as Anderson, and is also one of the great writers of science fiction.
