You read some part of Tom Godwin's Ragnarok series. If it was really a "short story" that you read, then it would have to have been the novelette "Too Soon to Die" which originally appeared in Venture Science Fiction Magazine, March 1957 (available at the Internet Archive) and was reprinted in the anthologies Science Fiction A to Z and Space Wars. Here is the editorial blurb from the original magazine publication:
The monstrous Gerns had left them to die—
left them on a hell-world of gigantic wolf-like prowlers,
of 1.5 gravity that dragged at aching bodies, and seasons
that withered life or froze it. It was unthinkable that
they could strike back at the Gerns 200 lightyears away—
it was impossible that they could even survive . . . impossible
The novelette was expanded into a novel called The Survivors (alternative title Space Prison) which is available at Project Gutenberg. (There was also a sequel called The Space Barbarians.) The Survivors has its own Wikipedia page, and on this site it was the subject of this old question and this one.
In fact I think you must have read the novel (The Survivors aka Space Prison by Tom Godwin), because the alliance with the prowlers and the telepathic squirrel-like creatures do not occur in the novelette, they are from the novel.
The "saber-tooth cat-type animals" are called prowlers:
They were things that might have been half wolf, half tiger; each of them three hundred pounds of incredible ferocity with eyes blazing like yellow fire in their white-fanged tiger-wolf faces.
The alliance with the prowlers:
The golden light of the new day shone on them, on his daughter and grandson and the prowler pups, and in it he saw the bright omen for the future.
His own role was nearing its end but he had seen the people of Ragnarok conquer their environment in so far as Big Winter would ever let it be conquered. The last generation was being born, the generation that would meet the Gerns, and now they would have their final ally. Perhaps it would be Johnny who led them on that day, as the omen seemed to prophesy.
He was the son of a line of leaders, born to a mother who had fought and killed a unicorn. He had gone hungry to share what little he had with the young of Ragnarok's most proud and savage species and Fenrir and Sigyn would fight beside him on the day he led the forces of the hell-world in the battle with the Gerns who thought they were gods.
The telepathic critters are called mockers:
They emerged into full view; six little animals the size of squirrels, each of them a different color. They walked on short hind legs like miniature bears and the dark eyes in the bear-chipmunk faces were fixed on him with intense interest.
[. . . .]
"They're telepathic between one another," he said. "The yellow one there repeated what the one you spoke to heard you say and it repeated what the yellow one heard me say. It has to be telepathy between them."
[. . . .]
Their intelligence was surprising and they seemed to be partially receptive to human thoughts, as Bill Humbolt had written. By the end of the fifteenth year their training had reached such a stage of perfection that a mocker would transmit or not transmit with only the unspoken thought of its master to tell it which it should be. In addition, they would transmit the message to whichever mocker their master's thought directed. Presumably all mockers received the message but only the mocker to whom it was addressed would repeat it aloud.