My advise would be to sell the original and use that money to buy an eBay one and get a garage to put some protection on it.
I'd have thought advertising expensive original Cat converters on ebay would have a downside. Some new thieves might realise just how much money can be made by stealing them (or cause damage trying) .
You could get a visit from police enquiring where you got it from (probably not these days
) or irate Jazz owners, convinced you are selling the one that was stolen from their car.
I dont think later cars are particularly at risk and this will be reflected in insurance premiums. My advice would be to make sure you have good insurance cover , and dont tell them the car is normally garaged overnight if its not.