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Not sure how much of a pain extension the roof (rather, just the tiles) a but would be.
As to removing render - why bother? And you certainly wouldn't need to render over the insulation in lime. And good luck finding someone that can do lime render and insulation.
If it's inside you want, you would have to account for where the vapour boundary would land - you could end up with everything condensating between the insulation and outer wall.
Stud walls with air gaps would mitigate that, but you'd lose space.
NotThamesWater-
As to removing render - why bother?
It's often shit, and ugly, and stops solid walls from breathing. There's no way we're keeping the current stuff we've got on the front of our place.
And you certainly wouldn't need to render over the insulation in lime.
I've lost track of what the original scenario @apc was asking about, but they might if they have a house built before the 1930's. It's quite common to need to render over external insulation and if your house has solid brick walls you need to use something breathable.
To reinforce what you said about EWI vs IWI and the risks of condensation and damp: Yes EWI is more expensive and harder to fit, but if you don't get IWI exactly right you can easily end up with a damp or mouldy house.
With proper EWI you get higher heat flow through the wall which helps it dry out. Walls often dry towards the interior, so if you stick impervious/non-breathable insulation on the inside you end up with damper walls and any timber in those walls will also be in a damper environment.
IWI does the opposite - it lowers heat flow through the wall, reducing its ability to dry, which means you're more reliant on external finishes keeping the rain out of the walls (whereas a proper EWI system will include something that does this, then any moisture in the walls is free to move into the interior of the building, which might sound like a bad thing, but isn't). This is true even if you install a properly detailed, breathable system using hygroscopic insulation with good capillarity.
Fox
I've considered it but how do you solve the issue of the roof not overhanging the gable end wall enough already, let alone once you add further insulation.
We'd need to remove all render (cement), add insulation, render with lime, amend roof, sort gutters etc which is why internal is appealing as we're gutting everything anyway