For all the non-rotating† parts of a bicycle, weight limit should be understood relative to the intended use case: I wouldn't trust the fork on a cheap big-box bike to do more than just carrying a normal-size rider, but on the other extreme a downhill fork that's intended for a 100 kg rider is actually expected to withstand more like 300 kg, because hard landings are part of the intended use case. A good MTB fork would definitely be capable of supporting your weight. If it sags too much, you can pump it to higher pressure.
What it may not be able to do is survive the kind of terrain that a lighter rider would use such a fork for... but I suppose you don't plan to do that anyway.
As for carbon fibre, the question can't be answered as such. Carbon is just a material. There are all kinds of different carbon fibre products, ranging from ultralight parts that wouldn't survive so much as a solid knuckle tap, to crash cells designed to save a racecar driver impacting a wall at 300 km/h. Cf. Do carbon MTB frames crack easily?
Good carbon forks are quite sturdy, but most of them are primarily designed to be light, meaning they will generally be weaker than a decent aluminium fork (although aluminium is a much weaker material, but thicker walls compensate for this). It's true that many carbon forks are also a bit more flexible than forks made from other materials, but don't expect too much: they can only filter out very small bumps, the kind that can also be dealt with by wide tyres at lower pressure.
I daresay neither a suspension fork nor a carbon fork makes a lot of sense for you coming from the point of view that it should withstand an unusually heavy rider. The kinds of suspension forks that might be useful are expensive and require regular maintenance, and that's only really worthwhile if you intend to ride hard off-road. Carbon forks are also expensive and useless for the opposite reason; their main benefit of low weight is quite irrelevant for you.
Instead, the most sensible option is a good-quality rigid aluminium or steel fork. Models designed for touring should be just fine with carrying your body weight, and suitably wide tyres will deal with most that a normal road throws at you. For the occasional larger bump in the road, you should not rely on suspension but instead get used to briefly standing up on the pedals and use the legs as suspension.
If you're actually interested in mountainbiking, then suspension does of course make sense – but then you should really get a whole bike for that purpose, not just swap out a fork on a bike whose rear end isn't up for that task.
†I'm saying non-rotating here because for wheels, every revolution is a fatigue cycle and it's not possible to reduce this aspect by gentle riding. Even a downhill wheel may not survive long distances at twice the intended load, though it will be more robust to potholes or lateral hits.
