All you really need to determine is the lowest minima which you can use to return for a landing with one engine inoperative. Starting with the CAT III case:
If the airport has an available CAT III runway, with all of its equipment functioning, and your airplane is certificated for a CAT III autolanding with one engine inoperative, and your operating specifications or manuals allow this, and you and the crew are certified for such...then no takeoff alternate is required.
But, if any of those conditions cannot be met...such as, the airplane is certificated for CAT III OEI but today it has a deferred item that precludes CAT III ops, or the runway is certified, the approach plate says CAT III, but there is a NOTAM downgrading the localizer transmitter...then you need a takeoff alternate.
If your airplane is only certificated for a CAT I landing with one engine inoperative, or if the runway is only certified for CAT I operations, then that's the weather you need to avoid a takeoff alternate. Or, if your airplane is certificated for a CAT III but your company policy doesn't allow this, or the runway is downgraded, etc., etc. This can get tricky with some airplanes. The B757/767 fleet are almost universally certificated for an autolanding with one engine inoperative as long as the engine fails following selection of landing flaps. This is good to know but it doesn't count for planning purpose; these aircraft are limited to CAT I (or CAT II if manual landings are permitted) for the one engine inoperative planning requirement (such as a takeoff alternate). Only some 757/767 aircraft are certificated for an autolanding if the engine fails en route, and these aircraft, all else allowing, could plan for a CAT III autolanding with one engine inop.
At the end of the day, just figure out what the lowest minima is for a one engine inoperative approach, at that airport, that runway, with those NOTAMs and that MEL. If you have it, you're good. If not, tack on the alternate. There is probably no less expensive operational change to make...it's not as if you have to add fuel!