Firstly, as others have already pointed out, you need to establish whether or not the exhaust is actually causing the rattle, or whether it is simply the heat shield plates.
If you do need a section of new exhaust welded in, you might want to consider using Infinity Exhausts:
At Infinity we custom build exhaust systems to your specification from only the best T304 stainless steel which we have been doing since 2002.
www.infinity-exhausts.co.uk
They build custom exhausts to a very high standard, and a couple of years ago I drove over from Norwich because they were the closest company that gave me that quality. Their silencers contain no internal baffles whatever.
In 2018 they quoted me £350 for a cat-back exhaust, or £700 for a full manifold-back system. I went for the latter and couldn't be more pleased with the result. It took just under 4 hours to fit, is really quiet, and the engine just runs so much better.
Beyond checking that they're not leaking, an MoT test tells you nothing whatever about the state of the exhaust. My own exhaust was showing the classic signs of a system that is becoming choked with internal rust, with very poor low end torque a particular issue.
It's not a great idea to patch in really short sections of exhaust, but a cat-back replacement stopping short of the rear box is more than realistic. Obviously in the current environment they may not be taking on work, but it's easy enough to ring them and get a quote.