I would just like to echo what Pete has said, contact Trading Standards, explain the background, and tell them you you believe the goods supplied were not of "
merchantable quality" and see what they think.
Not sure how the earlier refusal/repair situation will complicate this case, but worth a try.
If they give you any indication that they think you have a case, you can go back to Two Wheels and/or Speedway and tell them Trading Standards think you have a strong case for the goods not being of Merchantable Quality, and you will progress the case further with them unless they reconsider.
Most companies do not like getting hassle from TS, so may start to bend a bit. They may do nothing, but may make you an offer to cover some if not all of the costs up front. May be worthwhile taking a good offer, if you get one, otherwise it's back to Trading Standards to see how far they will go with it. Has to be worth a try as initially it just costs you the phone call or a trip to town to the local TS office.
Last time I used them for something, they were very helpful to me.
My situation was very different but got me my Alfa 156 it a very silly price.
The car was advertised by a large multi site dealership as reduced in price, but it was at a really silly price and I bought, signed up and paid for it, as it was just a bargain I could not refuse, even though I knew something was wrong with the price, but it HPI'd OK.
They then came back and told me the price was a complete mistake, the girl in the office had seriously cocked up, it should have been £4.5 k more, (but had been advertised this way for over a week). TS told me they had not got a leg to stand on, as it had been advertised on 3 web sites and on the forecourt at that price and I had legally completed the sale, so they could not do a thing.
The company got nastier, saying the price was obviously wrong I had to have known that, then I explained what TS had said and they got a bit nicer, "can we do a deal 50/50" etc, then after a few days it became "would you like to make some sort of a contribution as we are losing a lot of money".
I was almost in tears by this time as I was feeling so guilty

, but I just said sorry no, I bought in good faith. So I ended up with an 8 month old top of the range Alfa 156 for £10.5 K rather than the price it should have been on sale at, £15k, (around £24k new in that spec.!).
I still can't live with the guilt even now, almost 4 years later
Seriously though, I would have been much more likely to have given in to a compromise without TS backing.
Also turned out to be the best car I've ever had !