Short answer: As snailplane said, 器 was originally written with a 犬, and that's still the preferred form in China. Nowadays 大 is the standard in Japan, but you'll spot the older version from time to time depending on the font, application etc.
Long answer:
Ok so first, it's important to understand that RTK is not an analysis of how are kanji built, or what's the actual logic behind their components and how they combine. RTK a system of arbitrary mnemonics to help memorizing kanji (in the same spirit as similar mnemonic techniques to memorize an entire deck of playing cards, or digits of pi and so on). The stories in RTK do not necessarily bear any relationship to the actual reality of how were the characters made, and why. If they did, they'd be a lot messier, with lots of multiple meanings and graphical variants and characters based on pronunciation, etc.; and "messy" don't work very well for mnemonics. That's why the author decided to just invent "stories" out of the blue for his simplified system of "radicals". The system of meanings described in RTK is an invention of Heisig's; it's only loosely related (at best) to the system of meanings of the kanji themselves.
器 is a good example. In the modern Japanese shape, the middle part is drawn like 大, not like 犬. So if you're making stories to memorize Japanese kanji, you'd want to use radical "big", not radical "dog". However, the original shape of the character was something like this:
Now, compare it with the original form of 大, which was a stick figure of a person:
And compare it also with the original form of 犬, which was a drawing of a doggo:
It should be clear that 器 was definitely built from 犬, not 大. You can clearly see that slanted stroke standing for its head, which would later become the drop in 犬 when calligraphers simplified the shape.
Why does the standard shape in Japan use 大, then? Purely for convenience! It's one less stroke to draw, and it won't be confused with any other character anyway. This kind of simplification dates a long way back; even in ancient China, by the time 犬 started looking like 大, they would sometimes draw the doggie head in 器:
And sometimes they'd omit it and just draw the 大 shape:
Or even simplify it even further:
This latter abbreviation, 噐, with a 工 shape, didn't catch on enough to be widely used today; but 器 with 大 did, to the point of becoming the standard in Japan. Here's an image directly from the Jōyō Kanji table, the document in which the Ministry of Education defines the kanji used in the school system:
The first shape is the one currently prescribed for general use. The one in parenthesis is the 旧字体{きゅうじたい} ("old") variant. These were never different characters; the were just alternative shapes of the same character (like the 'a' and 'ɑ' characters when writing English by hand). So you were right: as it regards meaning, it doesn't really make a difference which one you use. You said that this variance doesn't make sense to you. That's probably because you've become used to Heisig's neat little system where every shape has only one meaning and all compositions always have a logic behind them. As we've seen with this example, the real world of kanji isn't like that. In particular, 大 in kanji combinations is often not a real 大 but a simplification of 犬, just like 王 in combinations is often not about kings, but a simplification of 玉.
Bonus: Why did they put a dog in there, anyway? The four boxes are clearly 'vessels, receptacles'; but what do 'dogs' have to do with anything? We don't know! The oldest explanation is that of the Shuowen dictionary, written in the 2nd century, which says it was a guard dog protecting wares. However, the Shuowen was written a good thousand years after the invention of Chinese writing, and today we know that many of its explanations were wrong. The "guard dog" theory might be a stretch. What is it that it adds to the meaning compared to 品? The respected Japanese kanji researcher Tōdō Akiyasu thinks 犬 had a secondary nuance suggesting "varied; of various kinds", by its association with dog races; that feels like even more of a stretch to me, but who knows? Whoever created the characters left no explanation of what were they thinking. Welcome to the world of kanji research.
(All images from chineseetymology.org.)
