If the coffee table is assembled then you can’t do the work with the tools discussed without disassembling the table. You might be able to use a bull nose shoulder plane with a temporary guide strip but you still have to do some chisel work and would just have to use your skills to control the depth. A possible option for controlling the depth would be to clamp a depth control strip onto the plane.
While I would think that it is technically possible for the work to be done on disassembled parts with a plow plane and the right sized cutter, that wouldn’t be the preferred tool. I would go with a rabbet (rebate in Great Britain) plane and a wooden one or a metal one would work if you know how to control the work. The wooden ones tend to be cheaper but generally don’t have fences or stops. The ones that do are called fillister planes and tend to be more pricy. The metal fillister (often commonly called rabbet planes) tend to be the best tools for this particular work in my opinion and can be had new for about $70 and used for $25-$40. Pictures would help.