Teshoo,
1. Under the Consumer Rights Act 2015, the Motor Trader who sold you the car is responsible for correcting and paying for any significant problems which occur within 6 months of purchase.
2. The 6 month Warranty you bought is worthless and unecessary and has been mis-sold to you and you should ask for your money back.
3. Ask your nearest Lexus Dealer to identify the problem and detail its resolution including cost, in writing, on its official Letter Heading, to you.
If the total cost is less than £10.000, then
4. Take the letter and show it to the Dealer who sold you the car and demand that he corrects the matter which has been identified by the Lexus Dealer. If necessary remind the Dealer that you have rights under the Consumer Rights Act 2015.
5. If the Dealer refuses to honour his Statutory Responsibilities inform him in writing by Recorded Delivery (and keep a copy) that you will have the problem corrected by a Lexus Dealer and you will sue the Motor Trader via the Small Claims Track of the County Court for the recovery of all of your costs including expenses.
6. If the cost under 3. above exceeds £10,000, then seek the advice of a Solicitor and ask the Motor Trader for all of your money beck and return the car to him.
Good luck.