Not a dumb question, in fact a very important one. No, you cannot fire a 243 WSSM in an "243 Handi-Rifle". The 243 Handi's are chambered for 243 Winchester or as its sometimes referred to, the 243 WCF. While there may come a day when Handi's are chambered for the WSSM's, its a long way off since their (the WSSM Family of cartridges) pressure specs are well above the Handi design capability.
The single most important aspect of centerfire rifles beyond basic gun safety is to have complete and specific knowledge of your rifles chambering and the ammunition if is intended to fire. And it sounds like you get it since you asked the question.
Each rifle is chambered for a specific cartridge design. Certain cartridge designs are called by several names, adding to the confusion. Sharing a nominal bore diameter as do the 243 WSSM and the 243 WCF, is no indication of interchangeability.
Only reliable, documented interchangability charts should be used as the basis for ammo use. There are some clear examples of interchangeability and some clear examples of disasters that could happen. First, 38 Special and 357 Magnum are essentially the same cartridge design except for length. It is possible in most firearms to shoot 38 Special ammo in an 357 Magnum. The reverse is not true, and is generally not possible because the longer 357 will usually not chamber in a 38. And it could be very dangerous if a 357 round were to chamber in a weapon designed for 38 Special level pressures.
In the same bore diameter that started this discussion are the pair of cartridge names 244 Remington and 6mm Remington. Cartridge-wise between the two, they share the same case, bullet diameter and are interchangeable except that the rifle that use them generally have different twist rates in the barrel. As I recall, 244 Rem's are slower twist, and don't stabilize heavier bullets as well. So even thought they are nominally interchangeable, the performance of each is distinct.
