My Japanese training is very limited (I know the hiragana alphabet, but I'm sure my pronunciation is terrible), so I won't pretend any confidence there. Of course, we both agree that English "l", English "r", and the five Japanese "ra/la" sounds (ら, り, る, れ, ろ), are all different sounds. We all know it's a bad approximation to say that "ら" is "la" or "ra", but it's the best approximation available. (I'm told that whether it's more like "la" or like "ra" depends on where you are in Japan, to some degree). Indeed, this one-to-two mapping problem is so well-known, that there are demeaning "jokes" about saying "lice" when one means "rice", and so on.
Anyway, my point was this: your description of English articulation had some minor errors in it, which I tried to improve upon.
> If I pronounce la (laa) ... in English, my tongue ... at some point touches my gums behind my bottom teeth
That may be true for you, but this is not typical. Or at least, it's not the the "L" that's doing that. If you say "raaaaaaaaa" in English, you'll find that the "aaaaa" noise puts your tongue on the bottom of your mouth just as much as it does for "laaaaaaa".
And if you put your finger on your lower gums, you can learn to say "la lee loo lay low" without your tongue ever touching your finger, much less your gums. Just like how many English speakers curl their tongue to say English "ra ree roo ray roe" but they can learn to leave the tip of their tongue down (with practice).