Solution 1:

This is an old , classic film. It's older than I am.

In the film the Bow-Wow Hot-Dogs literally bark in Gus's stomach. Donald throws a bone out the front door and Gus's stomach leads Gus after it, having been stirred-up by Donald pretending to be a cat. The Bow-Wow Hot-Dogs seem to transubstantiate into something canine in the stomach.

There is generally only one joke on Donald Duck films: everything turns against Donald. Gus was out the front door chasing a bone and Donald has barred that door. Then there is a smashing sound, and, Gus re-arrives through the back door (not through the doorway but through the door). The film ends with Gus stationed in Donald's icebox. No rhyme nor reason involved; Donald is confounded, the point of his films.

Solution 2:

A joke never survives explanation, of course, but from a sociolinguistic perspective much can be said about the US English-speaking, barking hot dog.

First, honorable mentions:

  1. When near a female in heat, male dogs bark and whine (see especially the comments at the linked site).
  2. When grilled, some sausages make sounds that resemble the barking and whining mentioned in Item 1 (above).
  3. The dog in question may be barking hot, or it may be a barking hotdog.
  4. Barking dogs are now and have traditionally been used to chase away pesty geese — like Donald's Cousin Gus. See "How to Train Your Dog to Chase Geese", for example.
  5. The phrase 'barking hot dog' in one slang sense is another name for a boastful showoff; in another slang sense it refers to anybody or anything especially skilful and ostentatious:

hot dog, n., adj., and int.
orig. U.S.
A. n.
....
2. slang (orig. and chiefly U.S)
a. A flashy, ostentatiously successful person; a show-off. In later use also: a person proficient at a sport, etc.; esp. one who gives a flamboyant display of his or her skill.
....
B. adj.
N. Amer. slang. Of outstanding quality or merit; skilful; flashy, ostentatious.

OED

Note that the two slang senses in OED are first attested in 1894 and 1896, respectively.

The mass noun 'hot dog', meaning "sausage meat", is attested as early as 1884, and the meaning, loosely, "a type of sausage" is attested from 1892. Those dates bear on the apocryphal origin story mentioned in the following historical account.

Next, some history of the phrase:

The 'barking hot dog' was a...very minor...trope in popular newspapers in the early 20th century. Between 1916 and 1939 (the year of Donald's Cousin Gus), I found 13 appearances of the phrase in the Newspapers+ Publishers Extra corpus (pay to play). The uses were primarily in midwestern US newspapers, with Pennsylvania somewhat overrepresented; outliers showed up to the south and east.

If your first language is not US English, the nuances of the following representative uses may be somewhat or completely opaque. I've very sparingly glossed some of the more opaque language in 'editorial' comments.

Tonight the big white tents will be folded at the conclusion of the evening's show. The carnival will be packed away...and will be on its way to Washington, Pa. ... Along with the show will go the press agent, jolly Ed Salter, his dark brown valet, Zimuel, with his green suit and wide southern smile, the pop corn man and the ever barking "hot dog" man.

The Daily Republican (Monongahela, Pennsylvania), 03 Jun 1916, Page 2.

That the festival will be a big event is assured. It only remains for the days of the whining merry-go-round, the barking hot dog, the wail of the toy balloon and the inspiring music of the brass bands.

The Culver Citizen (Culver, Indiana), 12 Sep 1923, Page 1.

Practically admitting that The Tribune is in the best position to give most satisfactory service to fight fans...The Mirror today...makes a frantic effort to convince fans that they should go...stand in the middle of the street car tracks, try to make their ears compete with the noise of "barking hot dogs," [reference obscure, but possibly drivers honking automobile horns — ed.] and "warming up" flivver [a motor car — ed.] engines, to "hear" the fight returns.

The Warren Tribune (Warren, Pennsylvania), 22 Sep 1926, Page 4.

Preparations for the entertainment of one of the largest crowds ever to have assembled...are being made.... Sandwich men will be there; soft drink stands will be in operation; balloon peddlers and barking hot-dog purveyors will ply their trade.

McAllen Daily Press (McAllen, Texas), 08 Mar 1928, Page 1.

barking hot dog movie

The Times (Shreveport, Louisiana), 24 Oct 1928, Page 8.

The Muncie victory over Frankfort [this puns on 'Frankfurt', see following — ed.] has made it a case of too bad — much too bad — for the teams that encountered the "Hot Dogs" since the defeat at Muncie. The barking hot dogs have been cantering about the state and chasing everything that resembles a basketball opponent upon a tree.

The Times (Munster, Indiana), 23 Jan 1931, Page 22.

...in the past the catering had been none too good at the Cracker ball park. The peanuts could have been better. ... And the sandwiches were not enthusiastic sandwiches. They were usually dry and likely to cause ennui.
  The hot dogs were never vigorous barking hot dogs. They seemed to be listless, lukewarm and noncommunicative in their lather of mustard.

The Atlanta Constitution (Atlanta, Georgia), 31 Mar 1933, Page 13.

After the program steaming coffee and barking hot dogs were carried around the room by ladies of home economics.

Tunkhannock New Age (Tunkhannock, Pennsylvania), 17 Oct 1935, Page 2.

barking hot dog store

The Journal Times (Racine, Wisconsin), 11 Aug 1939, Page 8.

After the 13 aforementioned uses in papers from 1916 to 1939, sampled above, I only found 9 more unique uses, from 1942 to 1998, in the Newspapers+ corpus, which includes newspapers "from the 1700s – 2000s".

One of those later uses brought an apocryphal origin story to my attention:

Do you know by what name hot dogs were often called before 1901? Vendors at the ball parks called them "dachshund sausages." As the story goes, one day sportswriter Tad Dorgan dashed off a cartoon depicting barking hot dogs but because he could not spell the word dachshund, he labeled them "hot dogs." The new name stuck.

The Tribune (Coshocton, Ohio), 09 Jul 1986, Page 5.

A further search for "dachshund sausages", however, turned up no references to same before a reference in 1906, another in 1907, and none between that date and 1938.

Considering the much earlier origin of 'hot dog' meaning "sausage", and considering that the origin story, in various forms, didn't began appearing until the 1920s, whereupon it appeared first without the "barking" element, the story is undoubtedly fabulous.

Solution 3:

A dog barks at unwanted visitors (generally speaking). My guess is that a barking hot-dog is a kind of hot-dog which can help one keep unwelcome guests at bay.

Just like a real dog barks protecting the house and those who live in it, "the hot kind of dog" (hot-dog) is protective too.