> 「[一気]{いっき}にこれ[以上]{いじょう}[修行]{しゅぎょう}した<b>って</b>[意味]{いみ}はない<b>って</b>。[限界]{げんかい}までやったんだ。」
>
> "As far as I know, the second って means と言っている and is used to insist on what precedes it, like : "And I'm telling you that..." in English. Am I right here?"

Right.  The second って is quotative, implying "Here is what I want to say and I know what I'm talking about.".

> "As for the first って, I don't know if it means という, or even というのは"
>
> "Or if, as I think, it's part of たって meaning ても "

It is part of <b>たって</b>, which is the informal way of saying <b>ても or たとしても</b>.  These express a <b>non-resultative hypothetical condition</b>, which is why your TL:

> "I'm telling you that even if we trained more than that without pausing, it wouldn't make sense."  

could not be any better.

> "Is there a way to tell one from the other when a word like 意味 comes after?"

You would need to look at the whole phrase, not just the single word, following the って -- in this case, 「意味はない」.  If the phrase after the って seems contradictory to the phrase before, you have what I called the "non-resultative hypothetical condition" above.
In the sentence in question, the two phrases are "to train hard" and "it is nonsense".

I could, however, easily form a sentence containing a 「って意味」 in which って actually can be replaced by という rather than by ても or たとしても.

「その[単語]{たんご}に『[食]{た}べる』って意味はないよ。」= The word does not have the meaning of "to eat".

So, you would need to pay attention to the whole context as usual.