I agree Mr. Woodhouse is first gentleman of Highbury but it is well known that he has, like Mrs. Smith in S&S, set himself outside the normal functions of society for health reasons. I suggest it is also well known that Emma is the decision maker at Hartfield thus the Cole’s would consider her disposition towards them before sending an invitation. I am sure her behavior towards the Coles has been well within the bounds of propriety but her opinion that she is the only person willing to teach them their place in society suggests her demeanor has been cool, less gracious than the Westons and Mr. Knightley:

The Coles were very respectable in their way, but they ought to be taught that it was not for them to arrange the terms on which the superior families would visit them. This lesson, she very much feared, they would receive only from herself; she had little hope of Mr. Knightley, none of Mr. Weston. (25)

Such behavior would rightly signal to the Coles an invitation would be considered presumptuous and IMO that is why they did not send one—they knew it would not be welcome for the two reason Mrs. Weston gives: “I suppose they will not take the liberty with you; they know you do not dine out” (25). Had the Coles felt free to send an invitation to Hartfield they would have when they invited the others. In this scenario there are no expectations of attention on either side so the Coles are not acting improperly—the families are simply not at that level of familiarity. I find Emma’s discontent rather double-dealing. She thinks the Coles would be impertinent to proffer an invitation to Hartfield but then feels snubbed when they do not give the offense. The Coles can’t win in this situation.

I think the Cole’s late invitation for Hartfield is prompted and their excuse of the screen is a white lie in the service of politeness. After all they cannot say they did not feel at liberty before and according to Mr. Woodhouse “they are so very desirous to have dear Emma dine with them” (25) it seems almost an apology for the “perceived” slight. My reasons for thinking the invitation prompted are the circumstances surrounding its arrival are suspicious. It seems too convenient that the invitation to the party arrives the day after Emma was speaking of it with Frank Churchill and rather annoyed at the compliment of exclusion:

They had been speaking of it as they walked about Highbury the day before, and Frank Churchill had most earnestly lamented her absence. Might not the evening end in a dance? …The bare possibility of it acted as a further irritation on her spirits; and her being left in solitary grandeur, even supposing the omission to be intended as a compliment, was but poor comfort. (25)

I think Frank took it upon himself or asked his father to give a hint to Coles that Miss Woodhouse was inclined to accept an invitation. Then there is the lucky coincidence that the invitation arrives while the Westons are visiting Hartfield “to make Emma want their advice; and, which was still more lucky, she wanted exactly the advice they gave” (25). The Westons are able to quickly overcome the last remains of Emma’s snobbery that had not already succumbed to envy. Frank had raised the specter of dancing to Emma and then at the party the “proposal of dancing -- originating nobody exactly knew where -- was so effectually promoted by Mr. and Mrs. Cole” (26) that it appears to me either the work of providence or more likely that of Frank Churchill.