Hello cyber gypsy and welcome aboard Filipino UK

This is a very difficult issue and one that has put people's lives on hold many times.
I wish I had a solution for you but sadly there's not a legal one.

Did you already look at the webpage Apply for a certificate of entitlement
You'll see it's not a simple issue and it has a considerable processing time too.

I was surprised to learn that the Embassy refused to consider an ETD for your child.
It seems they have reversed the procedure put in place last year following the Passport Office and reported here

Here's an extract:-

Emergency travel documents for children
Parents or guardians of children living overseas who wish to travel may be able to apply for an emergency travel document in place of both new (first time) passport and passport renewals for their children.

Customers will need to provide the same information as when applying for a passport for a child if the child has not held a British passport before. In most cases, we will also ask customers to provide evidence that all those who have parental responsibility for the child agree to apply for an Emergency Travel Document for the purpose of the proposed travel, or provide evidence that they have sole responsibility for the child, e.g. a court order. Customers can then follow instructions to apply for an Emergency Travel Document.
I don't know if it's worth following up or not. Either with the FCO or your MP or both.

Probably not given the unhelpful response you already received. But nothing to lose IMO

I can't help wondering if the Passport Office has improved it's lead times since last year so maybe the 7-9 months we've seen in the past has been cut down to something less than the 6 months processing of the Certificate of Entitlement.

I do feel your frustrations at the extreme processing times.

I'm just brainstorming here and haven't thought it through properly but I wonder what prevents your child from applying for a visa under the Philippine passport/citizenship then applying for a UK passport once in UK ? Well assuming you could meet the increased income threshold.

Of course the obvious risk would be refusal.
Then what?

As I said just a brainstorm idea to open up some discussion.

Anybody ?