As Ricardo pointed out on Twitter yesterday, if the council were to go down this route and the Allam's were found guilty, they would automatically fail the fit and proper person test and be prevented from owning the club(a fraud conviction is one of the few things for which the fit and proper person test is actually effective).
But despite the public anger and the support the Council may get, I seriously doubt they have the stomach for a legal battle with the Allam's and I don't think they'll go any further than trying to throw a spanner in the works with letters to the Premier League(ironically, if nothing else, they'll work against the Allam's on the name change application, the arrogance and unreasonable nature of those letters will damage the credibility of the Allam's and further enforces the idea that it's all really about a spat with the Council).
I don't think it's a question of having the stomach for the fight, the Council have various legal duties and also an obligation to consider the bigger picture. Yesterday you doubted they'd do anything, but the letters show they've looked at a number of options. Behind the scenes, they will have looked at more.
If what they do scuppers the arena plans and results in the Airco going back to community use, while at the same time, the club is not thrown into even more disarray, I think that's favourable to them getting dragged into the Allams personal issues.
They have to consider the situation from several different angles.