I would check condition of cat on the inside could it possibly be blocked or damaged?
Check if it's an oe part or aftermarket one.
To check the inside it would need to be removed.
To reinstall possible needs some new gasket ect.
Remove the down pipe and then under the manifold or the manifold.
You would then be looking to see the condition of the inside honey comb if this is all melted and or blocked this would probably cause the over heat you described.
I'm not sure but you could knock out the center really not reccomending this but might work to get you by till you find replacement.
I was tempted to do this myself check it that is and then replace.
This may help you to remedy this issue until you get replacement.
Down side is you may not like the noise and I wouldn't reccomend running without the cat for emissions reasons.
Sometimes the honey comb can be melted and this would block the exhaust fumes exiting properly maybe it causes a back flow and when it gets really hot then it let's it all go at once in the form of blue smoke?
As far as using for a family car if its really loud it won't be an option for you but if you figure it's definitly the problem you can look into replacing or having the thing cleaned if that is something you would be prepared to have done.
Maybe are other ways to check sometimes the comb can be loose you you might hear it rattle when shaken.
Could also use a bore scope camera type thing that would at least let you view with only taking off the down pipe.
This would cost something to get hold of but would make for quick diagnosis.
Limp mode can be annoying I have only seen it happen once and it was short journey.
It does help to save ur engine tho from catastrophic failure.
So does have that advantage can't really think of anything else other than saftey but that's kinda questionable ?