Hello Kev,
It can be done but it will take more than one season to kill-off the grass and weeds, otherwise you risk ongoing infestation in the garden bed = a royal pain in the arse.
I had a garden bed infested with Couch (Bermuda Grass), after years of fighting it, decided to cover the bed with chook manure and lime, water well, then cover with black builders plastic - just let it sit for two years and voila, no more grass but richer soil.
Perhaps better to
compost it separately, that way you make sure they're all gone and end up with a richer product.
Good luck.