It depends on how much you have to cut??.....but...
One option is to hire/borrow a petrol strimmer for the day (...weekend?) and effectively use it like the worlds smallest topper...although I wouldn't recommend leaving the cut grass to rot on the top as it has a tendency to kill whats underneath from light depravation etc, blitz it into mulch sized pieces and this is less likely to happen (..and the remaining grass can reabsorb the nitrogen as the mulched grass breaks down).....perhaps by strimming then passing the mower over the top, even though your's isn't a mulching mower by definition, it should still shorten the cut length :-k
Similarly, there are other animals besides sheep that could possibly manage it for this year...there are various businesses and charities that cry out for overwintering for donkeys (..usually october till march) so you could possibly have a clean pasture by spring, they eat pretty much anything, can survive quite happily on poor grazing and they don't poach the ground up too much due to their tiny hooves...hunting the internet would perhaps be the best idea for that one. :)
Or....sticking with the donkey theory, buy your own!...My sheep run all year with a donkey and they provide good comapny for each other, and as much as my sheep will eat grass of all kinds it's only the donkey that will deal with the docks and even the young thistles...the result, a pretty reasonably well maintained field, that hasn't needed topping or spraying for the past three years. :D
