What would you do?

I posted about this a week or so ago, had another look at my seed beds today - bl*%%y grass!!
I can hoe out the weeds and hand weed around my teeny carrots & beetroot because they're easy to spot - have given up all hope on the chard, salsify & snips.
Everything is in 2 beds about 10foot by 3, covered with debris netting on water pipe hoops & bamboo, weighted down with wood battens on the edge of the netting. it looks very nice, but it's a pain to get to for weeding.
Even if I pull out the loose weeds, and I removed as much of the roots as I could when prepping the beds, all that's happening now is that all the bits of grass seed that were left on the soil for 2 years are now germinating and i can't easily hoe what's looking increasingly like a lawn!
So should I just hoe what I can, leave the grass for now & hope survival of the fittest kicks in (in which cas I think the grass will win) or - glyphosate it, cover it & leave it til next year  or fork it over again to remove as much of the grass ( and seedlings    ), and replant minus the hoops so I can nuke the weeds & grass as soon as they appear this time - or is there any other alternative so I can have a few carrots & snips later this year?

Its a real pain when you cant get down to the plot as much as you want to to hand weed. I gave my beds a good hoe and hand on Sunday,but when I get down on Friday (weather permitting) I bet they have all shot up again....including some grass. The best thing to do is just keep on top of it as much as you can

Thanks for your encouragement all! I suppose what is nagging is that I am slap bang next to the row rep who has a beautifully kept plot, dead straight rows, not a weed in sight - they wouldn't dare    I will just have to keep looking to the left at my neighbour's plot which is more like I think mine will be - a mixture of some bang up to date weeded, partially ignored and abandoned areas    Someone please tell me spuds don't need weeding :blink:  Strangely enough, no one else on my side of the row is using any netting other than on soft fruit, and I have been told several times that there isn't really a problem with the dreaded fly, so perhaps I'll save the netting for me brassicas!

Rather than use a sprayer get a dribble bar to fit on to your watering can. This is a T pipe with small holes which allows you apply Roundup carefully to small areas without the risk of drift. Available and cheap from most garden centres.