You can remove the drawer and inspect the bottom board… if the chest is fairly new, it will be made of particle board if older it'll be a single piece of laminated or plywood… in either case inspect the back of the drawer where the bottom fits in. Often, there's just a retainer strip nailed or bradded in to hold the bottom once it's slipped into it's retaining groove in the other sides. Removing that strip will let you slide the board out and replace it with one someone will cut to fit for you at a DIY store (at least here in the U.S.)
Alternately, the bottom boards of drawers are often unfinished and often are made of cedar plywood to add a cedar smell (and bug protection). It's likely the perfume was only absorbed by the bottom board. Find a good quality shellac … has to be labeled shellac, not varnish or polyurethane. Give the drawer bottom at least two, if not three good even coats, letting it dry completely between coats. Shellac will seal the wood and prevent odors whereas the other coating will not…
Good luck!