Found this on the Bickersons website (http://bickersons.com/store/product11.html), regarding one of the DVDs:
Look closely! Do you recognize the set as the same one used on the earlier episodes of The Honeymooners? It is!
Can anyone confirm this? I've seen a few of the episodes from this collection, but don't recall the sets looking the same.

After some research, it looks like that the show in which "The Bickersons" took place, first appeared on series called "Star Time" produced at Dumont 1950-1951. "The Bickersons" was a summer series on that show lasting 13 sketches the summer of 1951. This would be prior to the Gleason Cavalcade episodes. So yes, it could be possible & would be interesting to see if Gleason used "The Bickersons" set & this is where it all started.

Frances Langford was the star of both "Star Time" & "The Bickersons" sketches. She was friends with Gleason & appeared a # of times on Gleason's show including "The Honeymooners" episode "Christmas Party" 1953.

Here is some links about the show, but no pictures showing the Honeymooners set....

http://www.tvparty.com/50-muir.html

http://www.imdb.com/title/tt0042147/?ref_=fn_al_tt_2

I always found it interesting that Frances Langford did that guest spot, seeing as how The Honeymooners was modeled after The Bickersons. Gleason was actually taken to court over the similarities, as detailed in this book (http://bickersons.com/store/bicbio.html). Ironically, Gleason himself considered legal action against the creators of both The Honey-Mousers and The Flintstones because of the similarities.

I took a look at The Bickerson's. Saw 3 of the 5 complete episodes that still exist from the 1950-51 season as well as 3 shorter sketches from summer of 1951, all on DVD. Neither of these 2 versions of the show contained The Honeymooners set. From what I have read up on, these are the only 2 versions of the show that took place. Pics included.

Oh & by the way. You thought Ralph & Alice had it bad. John & Blanche's bedroom was in their living room.