I would check the ATIS before deciding where to divert to, not once committed to an approach. I would divert to the nearest suitable, not necessarily the nearest, which might be an airfield a little further away with an ILS with the weather you mention...
As for constant descent then missed approach or fly level at MDA, do what your IR instructor tells you to do. In my company (big jets) we fly the constant descent and then do a G/A at MDA +50ft as flying level at MDA and then making a dirty dive if you get the lights is guaranteeing an unstable approach. Can be done in light aircraft, but can still lead to some pretty unstable diving as the lights appear just infront and 700ft below!!
So do as trained to pass your IR and then do as your company advises once you are employed.
PP