I sympathise with you and offer you my solution I've used for my blue coal mine clay See what your builders merchants have in the way of old out of date gypsum plaster , find out the company name and contact thenm to find the gypsum content .
I have used four bags of Thistle finish on my front and back lawn areas ,, it helped crumb the clay nicely , pervious to using it last winter I incorporated several tonnes of well rotted strawed stable muck and horse muck with wood pulp beddings.quare metre and rototilled in ,then tilled four more times while drying out and then treated again wwith thistle and again four times rototilled .
The Thistle was put on at the rate of 10 ounces per square metre on each dressing session .
Gypsum does not change the pH of the soil much where as lime does.
Old plaster board is now a recycled commodity , it is ground up and sold at extortionate prices to farmers for just the purpose you want it for.
My lawns were sown on the first of April and has their first cuts on 10 May a week after rolling it in three different ways to compact it and take out any small humps etc.