No!
Whichever way you look at it you need a person who is authorized to solemnize a marriage.
This can be a registered solemnising officer, a Judge, Justice of the Peace, Priest, Minister of religion or even the local City Mayor.
The costs that are being bandied about here (P15,000 - P20,000) seem to me to be way way too high.
If that was a 'true cost' then no-one in the Philippines would ever get married.
I may be out of date, but I don't think a Judge can actually charge anything, but they are allowed to receive forms of ex gratia payment /gift payments.
Traditionally the money is placed inside an envelope and handed over.
You certainly don't have to accept exorbitant silly costs just because you are a rich foreigner. Especially when you are not expecting any special service such as expediting or weekend weddings.
My advice is to have your fiancee find a lower cost judge or other authorised official.
Costs are not fixed but you can expect to pay something between P1000 - P3000
Eventually the choice is yours. After all you WILL need someone.
Normally you can extend your visa at the airport upon arrival to 59 days.
Otherwise you need to go to an immigration office. I strongly suggest you extend your visa. If the planning for the wedding has all been done and all your ducks are well lined up then 21 day visa is possible. But in your case it seems a number of things still need to be firmed up so a little 'wriggle room' may be needed.