"Entering the UK as a dual national
Americans may be familiar with the requirement that US citizens need to enter and leave the States on an American passport, even if the citizen is a dual national. There is no such requirement for British citizens. However, if you choose to enter the UK on a non-British passport, you will be treated as a national of the country whose passport you present to the IO.
For some countries, such as those in the EEA, this will make little difference to whether your entry goes smoothly or not. For others, it can complicate things a bit, but this does not mean you cannot enter the country on another country’s passport. Just because you have naturalised as a British citizen does not mean you lose your permanent resident status. If, for instance, you need to travel after your citizenship ceremony, but before you can apply for a British passport, you are within your rights to use your foreign passport with your ILR stamp or an unexpired residency card.
Where you might run into problems is where you try to enter the UK with a passport without any leave to enter. In those cases, you would be treated like a normal passport holder from that country. So if your non-British passport allows for tourist visas to be issued at the port of entry, this means that even as a citizen, you might be asked to prove you plan to leave the UK at the end of the trip. One way around this is to apply for a certificate of entitlement to right of abode (ROA) in the UK.
For an example of problems, if you are a UK resident who holds an American passport, and you have a child who is born in the UK with British citizenship, and you try to travel on his or her American passport alone, you might run into problems re-entering the UK. Obviously your family is entering to live, yet your child has no right to enter or live according to his or her passport. Although this might be cleared up with producing a birth certificate, the hassle and the potential problems aren’t worth it. This is a totally different issue than the requirement for the US, and the two should not be confused.
However, for instance someone who say holds both French and British passports are under no obligation to produce a British passport at the border when entering or exiting the country.
Questions:
Why might someone want to travel on a non-British passport?
There are several. Some countries, like the US require people to enter with that country’s passport if they hold citizenship. Some people do not want to have to bother with keeping track of two or more passports during trips. For these people, using previously issued permanent residency stamps or cards or ROA stamps might be easier. Others who travel to volatile areas might want to keep the entry stamps to some countries in one passport and the others in the other. It goes without saying in these cases it would be safer to only have the one passport in your possession. Or others who have lost their British passport or have yet to apply might need to travel before a new one can be issued."
There it is in case the link still wont work. The bolded bit is the part im referring to. Thats probably why you had the problems you did but refusing the birth certificate is the part where the CBP officer was wrong.