I'm afraid that I'd opt for the former; it is the more usual collocation. (And the unlikely chance that someone would mistake it for another kind of screen is minimal in context.)
My cellphone screen is cracked. My TV screen is dirty. My kitchen door is broken.
This is how we usually deal with 'possession' by non-sentient objects.
This is simply not true in this particular case. Unlike the attributive nouns in the other examples you've given, the word 'mobile' is most commonly used as an adjective. Thus, when it's used as an attributive noun in phrases such as:
Well, if you work in a mobile phone shop, then you must know. I've never worked in a mobile shop. I've been teaching English for the past 20 years. However, you did manage to come up with a lot of examples-- that aren't used, I take it?-- using the attributive rather than the possessive.