They let me through, but border force said to me I need to prove in some way I am British next time.
Only a British passport or 'certificate of entitlement' is legally defined as conclusive for the purpose of establishing British citizenship when entering the UK.
Is there another way to add perhaps to the Italian passport that I'm British?
Unfortunately it is more expensive than a passport. You can apply for a 'certificate of entitlement', to go in your Italian passport, but this costs £550.
https://www.gov.uk/right-of-abode/apply-for-a-certificate-of-entitlement
Additionally, the Home Office has given a hint that they may be developing an electronic solution to this issue in their newest eVisa FAQ.
British passport holders and Irish citizens, including those who previously held an immigration status but have since acquired British citizenship, do not need to do anything. British citizens who do not have a British passport because they are dual nationals (excluding Irish passport holders) may need to take action in future regarding how they prove their right of abode. Updates on this will be provided in due course.
https://homeofficemedia.blog.gov.uk/2024/10/02/media-factsheet-evisas-2/
For now, I'd suggest renewing your passport is the best option. £88.50 for a 10 year passport is not that bad a deal, in my opinion.