Sorry I'm a bit late replying to this topic.
A 1933 burial would have been before the start of the welfare system in this country. So no applications to the Benefits Office as we know now it.
The family may have paid for her funeral (coffin, service etc) but not the expense of a private plot. Hence the 3rd class. Or the cost might have been paid by the parish or taken from personal finances.
Private burial plots in municipal cemeteries have always been considered expensive and out of financial reach of some families but that doesn't mean the deceased didn't have a decent funeral. The family would have scrimped and scraped the money together for that.
Today Kensal Green Cemetery (a privately run cemetery) charge either £11,000 or £15,000 depending on location for a new plot plus upto £1900 for digging the hole.
A non-resident of the LB Hammersmith & Fulham can expect to pay £17,000 to be buried at Fulham Palace Road.
You should be able to get all your information from the cemetery's office. Maybe the person you spoke to couldn't be as helpful as they should have been.
It is highly unlikely the funeral directors have the detail of information you seek after this length of time. Can you tell me the company name? The company may still exist but could have been taken over by a larger group.
Even now, my father raises invoices for a funeral to the family and it's the family's responsibilty to take the bill to the bank or benefits office if they don't have that sort of money. If the family are benefit savvy, they will know upfront what the state will pay for and what it won't and family is liable for the shortfall.