I think you have a few options, not all of them are as good as each other!
Do you know anyone who could do a sketch of how it would look? You could get them to add in the improvements you'd make to the garden (weeding, possibly paving) to sweeten the deal.
You could swap your buggy for an umbrella fold side by side and get this type of storage unit (again, ideally with a picture to show how unobstrusive it is).
You could get a cheap double buggy with plastic wheels (like a city mini) and leave it folded and chained to the fence, covered with a travel bag or tarp. (Maybe if you did this with your current one and took a picture to compare to how nice the storage unit would look that would help your case!). This obviously isn't ideal!
You could get a light weight single buggy and a sling and use those instead of the double buggy – also not ideal, but at least you can get out of the house! (Could you do this just to walk to the car and transfer into your double buggy? Again, not ideal but might help temporarily.)
On a similar note, you could get two lightweight umbrella buggies and some of these straps to go between them www.amazon.co.uk/Prince-Lionheart-6550-Pushchair-Connectors/dp/B000056J23 – also not ideal!
You could offer to get a solicitor to draw up a contract saying you will remove the structure on the twin's 3rd birthday or something? Or at least offer to do this! Maybe get a nice bin storage unit and say on their 3rd birthday it will become communal property and his flat can use it for rubbish?
It sounds like an utter nightmare right now, I hope you figure out a way to live with it!