The law gives councils complete discretion over who they give DHPs to, how long they last, how much they are for, and what information the claimant must provide.
I wonder if the council are prepared to put all those conditions into writing so that there's proof of them. I think there's a very strong case that they breach Article 8 of the European Convention on Human Rights, the right to respect for private and family life. They also appear to infringe Article 1 of the First Protocol to the European Convention on Human Rights (at the same link as Article 8 - keep reading), the right to peaceful enjoyment of possessions. Breaching either of these Articles would make the conditions illegal via section 6 of the Human Rights Act 1998.
Councils have, unfortunately, been given far less money than they need to meet the needs of all affected by the 'bedroom tax', which is an invidious scheme by central Government to make councils look like the bad actors in this situation. It is understandable that councils want to try to maximise the effect of the meagre sums they can offer, but demanding scrutiny of your shopping bills seems to go far beyond the limited qualifications allowed in Article 8:
There shall be no interference by a public authority with the exercise of this right except such as is in accordance with the law and is necessary in a democratic society in the interests of ... the economic well-being of the country, ..., for the protection of health or morals, or for the protection of the rights and freedoms of others.
I've eliminated the qualifications that clearly don't apply. In truth, I can't see your shopping bills being of consequence to the economic well-being of the country, or the protection of health and morals.
That just leaves "protection of the rights and freedoms of others", which could be construed as allowing discretionary housing payments to be targeted to those in most need. That being the case, the restriction must be "in accordance with the law" and test for proportionality must be satisfied for the interference in the ECHR right to be legal.
I see nothing in the law giving the council the right to set limits on your expenditure (which I also believe amounts to a restriction on the peaceful enjoyment of your money, contrary to Article 1 Protocol 1).
I also doubt the requirements for proportionality are satisfied. The legal test is quite complex, but one key element is that the restrictions must go no further than necessary. I don't see how the council intruding into your private and family life by ongoing scrutiny of your supermarket bills now that they've awarded you DHP, or by setting limits on the pocket money you pay your son, are necessary to target the available DHP money appropriately. An example household budget for one week should be sufficient to see how much money you have left over after essential outgoings.
I am absolutely disgusted at the level of intrusion your council believes is acceptable. Unfortunately, it is difficult to frame a human rights argument neatly without legal assistance, but I've tried to give you an outline here.