It's probably going to come down to testing, because fatter tyres generally have a lower maximum pressure. I suggest starting with a wider tyre, and it's likely that the 29" MTB tyres are also ISO 622 (700c), so they will fit the rims. If they fit in the frame then they're likely to work better as long as they're rated to a decent pressure.
The equation you care about here is weight/pressure = area, in this case weight on the wheel divided by pressure in the tyre = area of contact patch. The bigger the contact patch the more the tyre flexes as you pedal, so the more you have to work. But a wider tyre gives a wider contact patch, so you get less flexing for the same area. A wider tyre is also deeper, so when you hit a pothole there's more give before you pinch flat.
Particularly on bad roads, and with difficulty ordering cool new things from your "local" bike shop,the usual solution of high-quality, high-pressure tyres is not going to work too well. If you could, I would suggest such a tyre, but on potholes and gravel roads those give a rough ride (meaning you also want a high-quality hand-built wheel).
One other tweak is to suspend the load somewhat. If you put a partly inflated tube folded up under the tool bag in the pannier the tools will bounce on that, reducing the stress on the tyre and wheel when you hit bumps. It will also grind against the sides of the pannier, so I'd put the whole thing in a sack. If that works (I've never tried it, but it sounds plausible), it won't help with the load directly, just the punctures and broken wheels.
From memory of a long time ago, I used to carry 50kg or so on a fairly stock MTB when I weighed 70kg, on fairly rough roads. But I also used to go through a rear wheel every year or two.
I'm wondering how much more I can safely carry on the bike without stressing it too much. The tyres seem to be under a lot of strain in particular. I'm 75 kg, unloaded with just me the bike tyres look fine, when loaded they're flattening out to an alarming amount.
