Well branno, as you've correctly guessed there's not much more you can do officially.

When you get married outside of UK you get a local marriage certificate, that's it.

There's no process to get a British marriage certificate, nor is there any process to actually formally register an overseas marriage in the UK.

Your also correct in that the best you can achieve is to have your original Philippine marriage documents deposited with the General Register Office. Documents will need to be translated if they’re not in English.
There is no requirement to do this.
Just means your marriage certificate is desposited and secured by the General Register Office as an official record and
makes it easy to obtain any future copies you might need.

Just be informed that you can't get the original certificates back again, once the certificates are deposited they can't be released again to you.

Any copies you request will be standard black and white certified copies of the original documents (and any translations) on official General Register Office paper.