I'd be leery of lining the bottom with plastic... landscape fabric, sure. But heavy plastic is going to create a "swimming pool" effect if you get heavy rains, and will also turn plant roots backwards when they reach it.
We use landscape fabric under ours (granted, ours are only 8" deep... if yours are 2 feet deep, it won't be as critical, although drainage may still be a problem) and it's often surprising how many plants will grow their roots through the fabric. Carrots do turn at right angles if they hit it (soil is so rich that carrots that are supposed to be relatively shorter rooted (Nantes type) often hit the fabric and turn 90 degrees, then grow another 4" longer! LOL!)
I would probably use the weeping hose on top, with a heavy mulch on top of that... that is how I set a couple up, and it worked well. Water naturally travels downward, and the effect is more natural than what you're considering. It's not that what you're doing won't work... it should, unless you get some really heavy rains. In some ways, you're creating a shallower "earth box" at ground level, but without the drainage provided by the perforated bottom of the Earthbox (google them if you haven't seen them, and even google "Earthbox Clone", for some homemade plans... you'll see how they work.
The thing about the Earthboxes is that if they get overfull of water (either because someone wasn't paying attention and overfilled them with the hose, or they were outdoors and got heavy rain) the reservoir simply drains out the overflow hole. Your raised beds won't have that ability.
Above all, if at all possible, use the best possible soil which is very high in organic matter. We filled ours with soil from our barnyards... over 50 years of heavy cattle (and horse) use meant it was essentially mostly well composed manure. Very similar in many ways to peat, it's almost black, super rich, and both holds moisture and drains very well. Even in hot dry weather (granted, we're not Arkansas here!) they rarely require watering more often than twice a week.
One other thing i'd consider, especially given your climate... "water holder crystals". These polymer crystals absorb water like a sponge (when full of water, they're like Jello... they can give you a very creepy feeling if you're weeding and you grab one in your fingers... you'll swear you've grabbed a really weird feeling slug!) They hold water after watering, allowing the soil to still retain oxygen (preventing waterlogging of the soil). They release it slowly as the plants need it, and repeat the cycle over and over. We have some large pots and planters that have had them for 3 years, and they're still visible and working. Here's one type (but scroll down for many more options, including some better prices, if you want to buy larger amounts)
https://www.amazon.com/Root-Natural...566843&sr=8-2&keywords=water+holding+crystals
They really can be lifesaving for plants in hot dry weather, and they've allowed me to grow in pots when I almost always lost my plants because we'd get busy haying or otherwise with the farm, and they wouldn't get the daily watering they needed.
Let us know how you end up doing this, and how it works. Always up for learning new ways of doing stuff!
Summerthyme
then the cloth ground cover fabric, a wonderful blend of clay *cough* and our own compost. Every year we added leaf mulch/compost in as well as coffee grinds, ground up egg shells, and even a little wood ash. Man oh man, I wish I could have brought all of that wonderful dirt down here.
Well worth the investment in time and expense. It is on our 'to do' list, but not the highest priority this spring. Hopefully we can get them built this fall to let the clay/dirt/compost blend over winter.