You can line the bottom with large concrete pavers, or pour concrete about 3" thick on the bottom. I think bricks might wiggle out of place, but if you've got a cheap supply, since those in the gravel as much as possible.
I've started doing dry-pour concrete for most of my pads. This would take a few days in your situation, (it can take a month if it's a car or tailer pad, since concrete needs to cure, regardless of how it got there.) But you just need it to solidify. I think the beds are deep enough that the alkalinity won't matter, it will be outweighed by the organic mix of the soil/compost/mulch.
Pour out the powder from the bag, keep it about 3 inches thick, level it with the back of a rake, then with a fuzzy paint roller and that gets it pretty even. Then, this is important, just mist (not sprinkle) the top layer with water from a hose or a mist sprayer until it's shiny. This creates a crust. slowly you will add water until it permeates the 3" thickness of powder. Then wait 20 minutes, mist it again. Every 20 minutes mist it until it's shiny for 5 or 6 times. Cover it, keep the sun off it, let it sit overnight, then sprinkle it with water from the hose at least 3 more times 20 minutes apart. If you can touch it with your finger and it doesn't give, let the water run gently all across the top until it's shiny. Repeat throughout the day several times, keep it covered. When it feels really solid in a few days after repeated soakings, then it's probably safe to add the soil.
I've done dry-pour for a car parking spot. I spent a month after spreading 4" of powder, putting wire mesh in the center of the powder, wetting it down a couple times a day, keeping it covered, made sure it was solid, starting to cure, and could support the weight of a vehicle. That was two years ago, and there's not a crack in it, it looks great.
the nice thing about dry pour is if you get interrupted it's no problem. It's not as heavy as wet concrete. You can walk on it gently if you need to, but it should be resmoothed if you do. I did a wide pad under an arbor in my garden and couldn't reach the backside of it, since it was next to a fence, but walking on it was fine.
Check it out on YouTube.