While Researching at HighWayCodeUK I came across this article from 24/10/2018, it is possible that minor changes to the requirments or otherwise have been changed since then, but it seems unlikely that it would be vastly different.
In this article, they mention Rule 57 of the Highway Code:
the highway code states: “When in a vehicle make sure dogs or other
animals are suitably restrained so they cannot distract you while you
are driving or injure you, or themselves if you stop quickly. A seat
belt harness, pet carrier, dog cage or dog guard are ways of
restraining animals in cars.”
As you said in you question, all pets must be "suitably restrained", underneath this are some examples of how to do this and I asume a carrier would be "suitably restraining your pet.
It seems that carrying your cat in a carrier is legal, although you may be accused of breaking Rule 57 of the highway code for having them on your lap. Although it seems unlikely to me that anything would be taken up on this. Even so my advice would be to put the cat in the back of the car in the carrier as penalties for breaking Rule 57 could be up to £5,000 and nine penalty points. I am led to believe that you can appeal these penalties if you feel you've been wrongly accused although I am unable to find anything specificaly for this Rule, but these are some common reasons for appeal:
- The contravention didn’t occur the charge is more than the relevant
- Amount a Fixed Penalty Notice had already been issued you are not the
- Registered owner/hirer the vehicle was being hired out to someone who
- Had signed a statement of liability it was being kept by a motor
- Trader the vehicle was being used without consent
I don't see how any of these relate to your cause, so it is probably better to just follow the Highway Code as best as you can where the guidelines are as vague as in this case.
I hope I answered you question, for more info I have attached a link to The Highway Code.
- Information about your question is on Page 22/23 (23/24 of the PDF)
- Information about appeals is on Page 27 of book two (101 of the PDF)
and Pages 76/77 (page 158/159 of the PDF).