I don't see why Charlotte would have said anything to dissuade Darcy from running over to the cottage to see if Lizzy required anything he could procure even if she should have been given a chance. First it would have to be assumed that Darcy and Charlotte could have a quiet conversation where she could contradict her husband safely without being overheard by Collins or Lady C or there would be hell to pay. Surely Charlotte would be sensible of that. There is no reason to suppose that Darcy would think that Charlotte needed secrecy to tell the truth. Secondly the woman who was wiley enough to meet her prey acidentally on purpose in the lane would have no reason to dissuade Darcy, of whom she has such hopes, from going to see after Lizzy. She would see it as a fortuitous circumstance.

I'm sure when Jane was sick Lizzy had just as much evidence that Mr. Bingley and Caroline were sensible as Darcy has that Charlotte is sensible, even though she did marry Collins. Yet no one questions Lizzy traipsing through the mud to check on her sister. Her doing so showed a pleasing affection for her sister. Darcy's going to check personally on Lizzy despite Charlotte's seeming sensibleness also shows a pleasing affection that will not be satisfied without firsthand knowledge that everything possible was being done for the invalid, possibly giving a few instructions to the servant for her comfort just for good measure. Certainly Darcy would know that just his inquiring after Lizzy would impress on the servant not to be lax in their duty.

I think you may not be giving enough credit to the concern of a person in love when they hear that the object of that love is unwell.

I do believe that Darcy was planning at some point to pop the question because we know that he had been so very careful at Netherfield not to raise her expectations unduly and now he was meeting her "accidentaly" in her walks. But I don't believe he went to the cottage with the intention to propose that night or even with the expectation that he would be let in to see her. He may, however, as he made his way to the cottage, have determined that he would propose to her as soon as she was well. Surprise.

]As long as he asked the question outside Mr Collins' hearing, Charlotte could give an honest assessment of Elizabeth's health. Now that I think about it, perhaps we should assume that Colonel Fitzwilliam asked such a question.