Really depends on who's doing the looking as to how obvious it is and why they're looking. A random visitor? Probably won't see anything. Someone who enjoys making such things (or stealing things) it doesn't take much of a discrepancy. I've built false bottoms, backs, tops into furniture many times for fun and non-profit.
Part of your problem will be that if your building with wood, too tight a fit will cause you problems when the humidity rises and ebbs. You'll want to use something that doesn't swell much for this, a hardwood plywood or HDF/MDF board.
You also need to match the inside of the locker with a similar wood or otherwise get rid of the bare wood. Scuffs and scratches in the paint and stain can also give it away which goes back to the too tight or too loose a fit.
The better option, if you were building from scratch, IMO is the bottom drops out of the locker rather than lifts out. In other words by pulling out some anchors you tilt the locker back and then the bottom falls out of it.
But given what you've got, it's not an option.
As far supporting the bottom, that can be easily done by simply using wooden blocks in the corners and usually you'll need at least one down the long sides and one in the center to avoid bowing and warping.
Item attachment is likewise easy enough I'd think using velcro as someone mentioned. Cover the bottom in strips of velcro, use more than you think you'll need. This will be pricey unless you find a great deal on it. Then just store cut pieces of the opposite side of the velcro in the space. A sheet of foam to drop on top of everything will help secure it down and also help reduce the thonk noise of a hollow space.
Opening the space is always an issue. Finger holes are a giveaway as are strings or anything attached. I would offer perhaps picking up a couple of welders magnetic angle holders, they're about $15 each at Northern and then on your false bottom attach a sheet of 1/4 mild steel, about $15 again at a big box store using duct strapping $2 for 10' at a big box store. Then to open you just empty the chest attach the welder magnet and pull.
Worst case you just turn the chest over and let the bottom fall out. Assuming you've got good attachments on the hidden goods they shouldn't go anywhere.
You may considering lining the chest with a tough non-marring cloth to help avoid any scratch or scrap marks. Or just scuffing the hell out of everything to start with.