The sefer Pidyon HaBen KeHilchaso 8:37 brings two customs. One is as you described.
The custom of starting the meal first and then performing the pidyon haben is brought by the Maharil, Shu"t Mahari Bruna § 121, Be'er HaGoleh 305:10, Migdal Oz 9:5, and Chochmas Adam 150:7, amongst others.
The Mahari Bruna gives the following reason for the custom: If the pidyon haben is done before the meal, there won't be a publicization of the pidyon, which is the whole point of the meal. It's also brought there to specifically break bread, do the pidyon, and only then continue the meal. The Mahari Bruna says this is to make the remainder of the meal part of the seudas mitzvah.
Pidyon HaBen KeHilchaso says there are more reasons brought in Likutei Pinchas § 53. There we find three other explanations (see there for the sources):
One is that the firstborns were saved from the Tenth Plague in Egypt at Midnight, while they were in the middle of their Pesach seder meal. Since they already started the meal, so too we start the meal before the pidyon haben, which is connected to the firstborns being saved.
Another is that the 10 Tribes sold Yosef after sitting down to eat a meal (Genesis 37:25), and the amount of money that a pidyon haben requires is the same as what they sold him for (ibid v. 28 says 20 silver coins, which is 5 selaim X 4 dinarim, see Rabbeinu Bachaye ad. loc.). Since pidyon haben is a redemption for their sin, and they sold him after they started their meal, we do the same.
A third explanation addresses your very issue: why is the meal only after a bris yet by pidyon haben the meal starts first. He says that a person can't eat the korbon Pesach until they first had a bris milah (Exodus 12:48). So the bris milah precedes the meal. However, the korbon Pesach has to be eaten before Midnight. Since Midnight is when the firstborns were saved (as stated above), and the meal was already finished, we start the meal before the pidyon haben. According to this, it would be best to finish the meal before the pidyon haben, but there's a concern people will get drunk and forget the laws. Therefore it was established to have the pidyon haben at the beginning of the meal.