British Embassy in Marshall Islands
There is no Embassy in the country and the Embassy in Fiji just as the British Embassy in Tonga does the work. Contact them for assistance when you are i the region. The Marshall Islands itself has a limited number of embassies and diplomatic missions abroad. As far as I know, the Marshall Islands does not have any embassies or diplomatic missions on its islands. The country has a limited number of embassies and diplomatic missions abroad, as well as permanent missions to international organizations.
Passports:
The High Commission cannot provide you with a new passport but can provide you with an emergency travel document. If you need to renew your British passport then this needs to be done online. You can see here how to do this.
Birth Registration:
The British High Commission does not do birth registrations any longer. You will now have to do this yourself by post. See the writing on British Born Abroad on how to register your child born abroad.