Elizabeth said:
That does put it in a different light. I was trying too frame it more literally, "although I say...." (now/for the time being, but that may change....)." In any event, it still seems both would be awkward in English for referring to natural processes, or really anything as obvious a dog getting larger over time.
Why? I'm still not really seeing the awkwardness, except maybe for the fact that we don't really have a "stock phrase" for this that's as commonly used as 「と言っても」 is in Japanese. Its usage really has nothing to do with the "process" of the dog getting larger and how "obvious" this is or isn't. It can be used whenever the speaker wants to qualify a previous statement.
Let me quote the two explanations (they're really basically the same thing) given in my 日本語文型辞典, maybe they'll make it more clear for you.
「前文で述べたことから期待されることがらに対して、実際はそれほど程度が重くないと修正をくわえ、限定するのに使う。」
例：
１）ビデオの作品を作った。といっても、せいぜい１０分の短い作品だけど。
２）仕事場が変わりました。といっても同じ階の端から端へ移っただけなんですけど。
「前に述べたことについて、それが実際にはそれほど程度の重いことではないとつけ足すのに用いる。」
例：
１）料理ができるといっても、卵焼きぐらいです。
２）シンガポールへ行ったといっても、実際は一日滞在しただけです。
３）去年は珍しく雪が降りました。いや、降ったといっても、ほんの少しで、すぐ消えてしまいました。
Does this make it any more clear?
Back to the dog example -- I have no idea what kind of dog it is, but clearly it's a breed that doesn't grow that big to begin with. So the speaker feels the need to qualify his statement that the dog has grown by pointing out that even still, the dog was small to begin with and so even now isn't
that big.
To make it completely clear while destroying any of the naturalness or brevity of the sentence (This is going to sound a lot like Nagashima-san's explanation in Japanese, but let me try to make it even more obvious)
"(My dog) has gotten so big now (compared to when I just got him and he was a tiny little puppy) -- well, though I say "big", I have to qualify this by letting you know that this dog -- being (a chihuahua or poodle or whatever) -- is small to begin with/by nature (i.e. isn't the type of dog who grows to be huge to begin with) and therefore is still relatively small -- or considerably smaller, at least, than you probably envisioned when I used the word "big" to describe it just now.