Sounds like nonsense. The dealer is probably incentivising staff for every finance deal they do.
You should be able to get discount and 0% finance. If not, walk away. As for it being illegal to pay cash plus a part exchange, that's what most people do anyway if I'm understanding you correctly.
Try another Mazda dealer, better still have a look on new car deals websites such as UKNewcars or Drivethedeal. These sites put you in touch with a dealer who is prepared to offer a discount, any deposit goes to the supplying dealer and you conclude your business with or without your part exchange with that dealer.
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