Here's the thing .... when grass grows through, it will pretty much 'lock it in place' ... but you probably don't want it 'locked in place' with the edges sticking up. So, once the grass grows through, it won't matter if the landscape pins are there or not. But, the grass won't pull the edges...
Keep in mind, over time grass will grow through the hardware cloth apron. It will be very difficult to pull up and get that extension cord out if you ever need to change it. It may be worth keeping it above ground and working through a seam in the cloths or cutting/patching a spot for it.
I was at Walmart over the weekend. This is what I had been using:
https://www.walmart.com/ip/seort/247378944
$0.09 ea. Drilling pennies might be better. Not sure how quickly that would wear out a drill bit.
How hard would it be to drill a hole in a nickel? You could have them for $0.05 a piece. If a penny is sufficient in size you could save another $0.04 each.
Be sure to take a look at this post by @21hens-incharge for a great way to tarp a hoop coop/run.
https://www.backyardchickens.com/threads/shower-curtain-vinyl-sheets-for-winter-winds-how-to-hang-how-to-secure.1547060/#post-26122184
They can also be purchased on Amazon...
Sometimes, in lieu of wood I used simpson strong tie straps and zip tied the hardware cloth to those.
https://www.homedepot.com/p/1-1-4-in-x-18-in-G90-20-Gauge-Strap-Tie-ULSTA18/313507640
It's probably the best option if you can swing it.
I like having pavers in front of my door, but they don't replace the apron but push it out a little more. I guess if you have enough pavers, you would eventually get to a point where predators would give up.
It took me MONTHS to get my coop/run done. But, it was mostly done an hour or so at a time in the evenings. I'd spend more time pulling out tools/supplies and putting things away than I did building.