As a fluent Hebrew speaker, I never really thought about this, but now that I do, it's true, the lamed is not pronounced strongly. However, it's noticeable, not least because lamed softens the following vowel (like in French), the way nun would not.
I wrote "we don't have air conditioner at home" and it was accepted. I think Ahmed's suggestion is rejected because בבית is "at home", and not "in the home" (which quite frankly sounds weird to me regardless)
I hear: איננו instead of אין לנו. No lamed. I agree with stan. Obviously, "we cannot AC at home" makes no sense, so there has to be an implied lamed. Thanks Ynhockey et al. for confirming our suspicions about the weak lamed.