As I understand it the problem is not with having two passports (which can be achieved fairly easily), but with using them. It's a kind of "Schroedinger's baby" scenario - a quantum paradox for the modern world if you will. The child has two citizenships - but as soon as an agent (border security) makes an inspection, both waveforms collapse and you are left with a singularity, or more likely a bawling child and a seriously harassed set of parents... Okay, I think I am getting a bit carried away here. Consider the scenarios though:

You try and leave China for a trip to England. At Entry/Exit, if you show the Chinese passport there will be a problem as there is no UK visa. So instead show the UK passport. This is now fine as obviously you don't need a visa for the UK. Except though that passport doesn't have a residence permit or entry stamp in it, so again there is going to be a problem for a different reason. Moreover, you can't just get a visa in the Chinese passport for travel to the UK, as once she has a UK passport the child is considered a UK citizen, and therefore not eligible for a visa... Okay, so get a residence stamp in the UK passport. But you can't do that as the PSB won't issue a residence stamp for a Chinese citizen. Checkmate.

From what I can gather the Certificate of Entitlement is one way around this, at least making the situation workable until the child is 18 (at which point they can legally renounce citizenship in most countries). On the Chinese side I think (and here the details are equally if not more hazy) there is a procedure to apply for a similar document (in exchange possibly for the Chinese passport) from the PSB, which can be used to exit/enter China (once only though - though you can apply repeatedly for this document) in combination with foreign passport. The whole thing though does seem a mess (and as LOUDNOISES notes even the advice on the British Embassy website is to apply for a visa in the of the child having a Chinese passport and doesn't mention the Certificate of Entitlement option).

With regard to the original question, I'm afraid I cannot offer any solid advice based on experience. My reading of the relevant text (from the UK Border Authority "Guide to the right of abode") is that the phrase "immigration stamps showing the passport holder is resident in the UK" just means "in the UK" - as this matches the tick-box on the application form (https://www.gov.uk/government/upload...0/form_roa.pdf) where you have to confirm you are in the country in which you are applying from.

You can certainly apply from Beijing using the online application form - and they will charge you an extra 140GBP or so for the privilege. That will mean though that at some point you will have to hand over your daughter's Chinese passport while the application is processed, with all the complications that entails regarding travel. Here you are best contacting the Embassy directly, as this is where you will most likely have to hand in the application.

As you will be in the UK anyway I think my approach would be to call the UK 0845 number (Skype is your friend here) and try and find out (i) can you apply for this for her given that she is in the UK on a visit; (ii) how long it should take, and (iii) how quickly and simply can get her passport back if the process takes too long and you all need to return to China. If you get positive sounding answers to all those questions then try it (in the worst case you will lose your 150GPB fee and you will have to apply again from in China). What you want to avoid of course is being stuck in the situation where where your wife has to return to China (due to visa expiry) while you remain holding the baby, both literally and figuratively...