I had a similar issue with the drawer under my gas cooktop. Actually, for a gas cooktop many people use a false-front, as you likely have here, for the top drawer, but most gas cooktops now are extremely low profile, so that the only issue is the gas supply pipe, similar to the issue here of a trap with a lower drawer.
My solution was to cut the back piece of the drawer, cut a rectangle out of the bottom of the drawer and cut some pieces to make a fence around the rectangle. You can use either wood painted white or melamine board (which is typically not wood but some sort of fiberboard/etc. but strong enough for cabinets/drawers).
| | Trap | |
| | | |
| +--------+ |