You can readily buy flush-mount pull rings for this purpose. Easy to rout the Starboard as said above.
FF mentioned the carriage bolt pull; works in solid surfaces as well as carpeted. Saw it on a Mainship 350/390 engine access door.
Easier, and cheaper than both the above is a finger hole, carefully eased, rounded, around the opening so not damaging to the finger. Presuming no stiletto heels, a 3/4" hole is not very obtrusive.
Similar finger holes in the Teak and Holly sole in our LeComte had neatly cut (radial slices not extending much past the hole) leather patches tacked on to the bottom side. The leather had enough stiffness to stay shut but a finger would readily poke through to grab the panel.