You can argue until you're blue in the face, this is not a decision that is based on logic or on the arguments you are provided. The 'arguments' and 'reasons' they give you are nothing but excuses. Even the best company will generally pay you the least amount they feel they can get away with. There may be some legitimate reasons why they won't give you the desired raise but in the large majority of cases they're simply not going to give you one as long as they think they can get away with it.
So how do you break through this pattern? The answer is that you have to be perceived as willing to leave over this. You don't have to use threatening language, simply stating your position will be enough.
I understand your reasons for denying me a raise but the simple fact is that if I don't get it, I'll have to start looking for a new job if I want to be able to keep providing for my family. I like working here because [reasons] but that's not as important as providing for my family is.
The answer to this is probably going to be some variation on "I'll see what we can do".
Be prepared for the answer to be "Sorry, nothing can be done, you're not getting a raise". Even if you do get the pay raise, be prepared to find a new job in the medium term, because squeezing the raise out of them when they really didn't want to give it to you usually raises negative emotions. While personal situations will often limit your options, they shouldn't be zero. You may be forced to take an undesirable one if you are to continue providing for your family.
Having said that, you'd be surprised at what suddenly becomes possible when an employee the company really wants to keep is threatening to leave. Frozen budgets can suddenly open up, or suddenly you might 'be able to get the raise if you keep your mouth shut about it because it might cause issues with morale'. If you do get a promise of a raise, get in writing how much of a raise you will be getting and starting when, then assess for yourself how likely you think it is that the promise will be kept or if it's just a tactic to keep you long enough for them to find a replacement willing to do the job for the money they're willing to pay.
If this seems cynical, remember that companies often feel that they are unique and that employees should be happy to get to work there. Management tends to feel that they know pretty well what people's services are worth and by providing those services at the price they are paying you and your colleagues are confirming those beliefs. Management can get so convinced of this belief that they'll let their best employees walk out the door or even fire them, thinking that they'll just find someone new or even someone better. It's only when the market proves them wrong that they'll grudgingly consider that they might have been wrong and even then, usually some poor fool will come along that is willing to take the offered pay, once more strengthening their belief.
