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I really, really, really would not do it on that rationale. (Although there are other perfectly good reasons a person might want to do it).
It is MUCH HARDER to predator-proof a run than a coop. Thus, you are taking a fairly simple project and making it fiendishly difficult and expensive, and much less likely to succeed.
I would sure not go with a mere 2' apron, I would do 4-5'.
Raccoons don't mostly dig under, they mostly climb *over*, as can foxes and dogs and other things; do not rely on electric to keep them out, as all electric fences WILL fail from time to time. So you would need either a very very strong and impenetrable wire top to the whole run -- I would suggest 2x4" welded wire mesh supported on strong 'rafters' -- or a solid roof.
(Another theoretical possibility might be to run 2'-wide metal flashing along the top 2' of your run, with only one seam and be very, very, very careful to construct that seam so it does not offer climbing handholds, and then put deer netting over the top of the run to exclude hawks/owls; but I have never seen or heard of the flashing idea being done so can't vouch for its usefulness, and it won't work at all in a very windy area or if there are overhanging trees).
And then you have weasels, which it will be essentially impossible to keep out of your run. And while you may not THINK you have weasels around, I would point out that most people don't think so, til they wake up to a massacre. (There are some weasel-free areas of the country; you could look it up and see if you happen to live in one. I wouldn't hold my breath)
Honestly, coops are pretty easy and inexpensive to predatorproof. If you simply don't want to be a slave to shutting the chickens in and letting them out at specific times, I'd suggest building a small run, as thoroughly predatorproofed as possible, for them to use as a 'porch' on days you can't let them out real early or have to shut them in too soon. Then have a larger, less secure yard for them the majority of the day.
JMHO, good luck and have fun,
Pat