Who the heck would want a GFF over a bungalow? 
Most bungalows are in nice areas (or okayish areas!) and many flats are in average or shitty areas. Often in high rise blocks - or 3 or 4 levels blocks. They often have loads of yobbos hanging around, and often have lots of young people and couples in them. You will have someone above, and the sound of people trundling up and down the stairs, and also the sound of the lifts! And in addition, many of them will not have parking closeby, and it can be several minutes walk.
Bungalows have a garden, they are often in residential areas, they have no-one above you, and are often in an area where there is other older people and disabled people, making it (usually) more quiet and calm. Not a guarantee! But more likely.
@Bungano have you tried either other social housing landlords (like other housing trusts and housing associations?) OR what about trying for a slightly obscure area... I know a couple who were in a 2 bed private rented house, and she had/has a chronic illness. Because they had a downstairs loo, they failed on medical grounds for a bungalow, but the council said 'you can have a flat though!'
They also said they were too young for a bungalow anyway (at 48!) So she registered with another borough, who had more rural properties that were hard to let because they had no public transport or shops or services.. doctors/dentist/pharmacy.
Within 6 months of registering, she was offered a 2 bed bungalow in a village. 63 properties in the whole village, and around 175 people. It has only a postbox, a Church, and a pub. No shops, no public transport, and no facilities.... The people who run the Church organise hobby groups and events, and it's a lovely little community.
Her and her DH said 'this will do for now' and planned on swapping/transferring to a busier, more convenient area, but within a few months, they loved it so much, they decided to stay. There is very little there and no public transport, but they are only 3 miles from the closest town.
They got it so quick because they were the only one bidding on that property that week. The properties in villages like this are hard to let, because there is very little there, but as I said, it's only 3 miles to town.
Would that be an option maybe??? It may not be ideal, but it would get you on the social housing ladder... If you find it too rural you can put in for a transfer.