When faced with a similar situation, many years back-
I’ve ‘tied up’ the ballcock(with my boot lace) – flushed the cistern repeatedly before pulling it all apart to replace the worn washer (Or flip it over and pack it out with blue tack.)
Just before the torrent engulf the overflows capacity I did drop the split pin pivot – so bent a galvanised nail in the hole – bodged yes -but sound for at least a couple of years.
In hindsight I’d only do it this way again, to a soundtrack of the ‘sabre dance’
If you have a modern plastic parts cistern- go easy with the mole grips.
edit – (If you turn on all the other clod water taps downstairs)plastic pomagne corks will jam into most old style taps seat whilst you fiddle with their washers- now who wants me to bleed their brakes?