The 'sole responsibility' rule can cause many applications to be very complex, as the meaning of the term cannot be precise. Every family’s situation is unique.

It has nothing to do with the Philippine Authorities, it's solely the British Embassy that you have to convince. You can have custody but not sole responsibility and you an have sole responsibility but not custody, but you have to convince the Embassy you have both.

Personally, I have not seen any written information from The UKBA on specifically how to prove Sole Responsibility status.

Remember that Sole Responsibility is not the same as sole custody. Sole Responsibility is a requirement of UKBA only.

Here's what UKBA say:-
What is sole responsibility?

You'll need to think carefully about the documentary evidence you need to satisfy the UKBA that you have been the main person exercising parental responsibility for your child by providing the large majority of financial and emotional support required.
You must clearly show, with examples, that you've had, and continue to have, care and control of your child.
Who has been entrusted with day to day care and control of the child since the sponsoring parent migrated ?

Proof of paying school and medical bills, whilst important, will not on it's own satisfy UKBA

The following factors should be considered when preparing documentary evidence to be used in assessing sole responsibility:-

How long has the sponsoring parent been separated from the child?
What were the arrangements for the care of the child before and after the sponsoring parent migrated?
If the sponsoring parent migrated to the UK, what has been / what is the sponsoring parent's relationship with the child?
Has the sponsoring parent consistently supported the child by:-
- direct personal care
- by regular and substantial financial remittances?, and in what proportions is the cost of the child's care and upbringing borne?
- by making the important decisions about the child's upbringing? (for example where the child lives, the choice of school, religious practice etc?)
- by enabling a significant degree of contact and communication.

The outcome of any Sole Responsibility centred application depends largely on its own particular facts and circumstances, and the ability of the sponsoring
parent to provide evidence.

In those cases where a child fails to qualify for a visa under the Sole Responsibility rule, there may alternatively be a case for entry to the UK on the
grounds that there are serious and compelling family, or other reasons, that make an exclusion from the UK undesirable and not in the best interests of the child.
However, the requirement of ‘serious and compelling family or other reasons’ may be difficult to satisfy.
It would not be sufficient to simply compare the harsh and poor conditions in the child’s country with better conditions in the UK.

Here's some reading that may help provide some background understanding :-

Children of British citizens and settled people

Children

Immigration Directorate Instructions Chapter 8 Children

There are quite a number of threads on this forum dealing with this topic so please do also make some searches.

Anyway Hopefully there's some food for thought.

Depending upon the complexity of your specific situation you may feel the need for professional input.
Think outside the box - include every instance where you have provided for the child and prove it. For example, a letter from the child's school stating that they deal with you only will help.