Hopefully the scenario below is not unique and someone can advise if this is possible:

I was looking at renewing my British passport, and applying for my kids, but noted that overseas applications are a lot more than when in the UK. We are all dual nationals so would it be possible to enter the Uk on our alternative passports, apply for new british passports in the UK and the have them sent to either our overseas address or a relative in the UK to the courier to us overseas?

I'm probably overlooking someone fundamental like being able to prove an address in the UK but to save a few quid I thought I'd investigate this option a bit further.

I don't see a problem with your plan. The passport office doesn't keep records of where you live nor will they know if you'll still be at the return address you give. I presume you've visited the attached website?
https://www.gov.uk/renew-adult-passport

don't know if it is possible but you could book the one day appt for when you are in Britain, I did that for my husband when he came home as his passport ran out sooner than he thought, it is more expensive but might be worth it