So, in the film, Gimli had around 5 or so axe props: two different sets of hand-/throwing-axe (one-handed) and bearded-/walking-axe (two-handed), both single-bladed, and also a two-handed, double-bladed axe. This last axe has been termed Balin's axe since it didn't appear in Gimli's hands until right after the cave troll smashes Balin's tomb. It was called Balin's Axe in the handheld LEGO LOTR videogame as well. Obviously Balin didn't have this axe in the Hobbit movie, although he may pick it up some time before the end of the third film (although some claim that he found it in Moria and it is actaully Durin's Axe). That is beside the point.
In the first wave of LEGO LOTR sets, Gimli is given a one-handed, single-bladed axe and the two-handed, double-bladed axe. Since both sets occur simultaneously to or after Moria, there is no problem here.
In the first wave of LEGO The Hobbit sets, Gloin is given the same 2 axes as Gimli from the first wave of LOTR sets. In The Hobbit: AUJ, Gloin is depicted as having one set of the same sets of one-handed and two-handed single-bladed axes. Since the set was based on events happening in the future The Hobbit: Desolation of Smaug film, however, it is possible Gloin gets his hands on a double-bladed axe of some sort in the second film.
In the second wave of LOTR sets, Gimli is depicted, from what I have seen in pictures, with only one axe. It appears to be a single bladed, two-handed axe on a stick that is longer than the wand/one-handed axe stick but shorter than the pole/staff/Balin's axe stick. This is in both the Council of Elrond set and the Pirate Ship Ambush set, so it is not only representing the axe that he breaks on the One Ring in the film.
I think Gloin ought to have been given this newest axe as well as with the shorter, single-handed one. That would have been more accurate to his movie depiction, rather than giving him the large, double-bladed axe.
I think Gimli's page ought to be changed to say he has three axes as his accessories, after it can be confirmed that there is a new axe with the release of the new sets.
Edit: The point I forgot to get across is this: the two axes (the one-handed and two-handed single-bladed axes) that Gloin carries in the Hobbit: AUJ are supposed to be hereditary axes that are passed down to Gimli before he leaves Rivendell with the Fellowship. The double-bladed axe that Gimli found in Balin's Tomb is not one of those axes and so it is incorrect for Lego to give it to Gloin as one of his accessories.
Balin had no axe, only a knife and a hammer/sword. Also the double-bladed one is not from Balin, it has Ereborian design, like Gimli's other axes. The only new axe Gimli got in Moria is a single-bladed 2 handed with Morian design.
Also, Glóin indeed never had a double-bladed axe, he had only 2 axes, a single-bladed short one and a single-bladed 2 handed one.
Here are Gimli's axes:
Council of Elrond Axe: short, double-bladed
Morian Axe: long, single-bladed
Double-bladed Axe: long, double-bladed (included in the sets)
Throwing Axe: short, single-bladed (included in the sets)
Note the picture of the five axes on the right hand side of the LOTR Wikia page for Gimli. Those are the five axe props that Gimli carried throughout the movie: A walking axe, a broad-/bearded-/battle- axe, two throwing axes (each matches one of the bigger axes), plus Balin's Axe from Moria. (Yes, I know Balin didn't have an axe in the movie, as I said before; he may get it in a future movie or he got it between the two series). The axe that Gimli breaks on the One Ring may not even be his axe (watch the scene again) but I am not counting it as one of his five axes. If you want to count it, then there are six axes.
Not only do others agree, I have watched the films recently and carefully to verify the Balin's Axe theory. Also, play the Moria level on the handheld version of LEGO LOTR. This supports the theory also.
I don't know how you are judging Morian design and Ereborian design so I won't argue that either way. These pictures show Gloins axes; it is easy to see which ones of Gimlis they match.
As for Balin's Axe: In the books, the weapons of the other dwarves besides Thorin are hardly mentioned. (Thorin has Orcrist, a bow from Beorn, and an axe from Smaug's hoard; the other dwarves have knives and some bows from Beorn, and their hoard weapons are not described; Dain II Ironfoot and his army wield mostly mattocks, like movie Bofur.) Thus, we don't know if the dwarves will lose their weapons permenently in the second movie. Likewise, the case of Balin's axe (as well as the weapons of all of the non-Thorin dwarves) is based purely on the movie evidence. Balin's axe may very well be from Smaug's hoard, created in Erebor after the Hobbit, or found in Moria when he tried to reclaim it (the reason why some fans believe it to be Durin's Axe).