Changing pesos to Euros or US$

Do it in US dollars.

What did they mean by "outside"? Maybe it just means you have to change your pesos to euros or US$ through money changers, and then deposit euros or dollars with HSBC.

I am pretty sure this is a new government regulation, as the company I use for transfers had PHP to AUD as an option 2 years ago, but they no longer show it.  When I asked about it, they said to check with them for each individual transfer.  But you need to check early in the day for PHP transactions.

My guess is they have an account in each country, and you make your local deposit to them, and they then convert it.

The current EUR-PHP rate with them is 51.12, not sure about the other way around.

You can contact the underground bank, convenient, but the exchange rate than the international exchange rate is low

Maybe you misunderstood something. It is not allowed to bring more than 10.000 Pesos out or in the country without declaring it.
It is also not allowed to bring out any money which you have NOT brought in. (proof of Import of the $)
It is also not allowed to earn money in the Philippines without paying taxes (Income Tax return)
Take your pick and ask yourself why your banker gave you the advise to rather change the money in the grey market.

Because Banks happily change any amount of money (and take the commission) from Peso to $, but they have to explain to the Central Bank where it came from and they are restricted a lot in doing so.

PS- In addition, if you are lucky to get the $ somehow- there is a Worldwide Law, that you always have to declare any amount exceeding 10.000 $ when travelling, regardless of in cash, Check, TC, bonds or other negotiable instruments. (Money Laundering Law) It is also written in all the customs forms on this planet and even sometimes in Tickets etc.
If you get caught, the cash is gone.

And for EURO Banking I found the Standard Chartered Bank to be the best so far. (others may also be there, just a personal experience)

Stranded Shipscook wrote:

It is not allowed to bring more than 10.000 Pesos out or in the country without declaring it.
...
there is a Worldwide Law, that you always have to declare any amount exceeding 10.000 $ when travelling,

That does, as you say, only apply when travelling and carrying on your person. When using a bank, they automatically notify the Anti-Money Laundering Council of any transaction over 400,000 pesos.

It does not apply when using a bank to do the transfer.  The OP appears to be having difficulty getting his Philippine bank to transfer it to his overseas bank.

No idea why the Philippines has that 10,000 pesos restriction, in or out. (US227/ A$285)
The figure for Australia is "AU$10,000 or more, or equivalent in foreign currency (meaning notes and coins of legal tender)"

10,000 pesos does not last long when you first arrive.