The Cambridge Dictionary is unfortunately behind the times here, which is important when we're discussing an example of informal youth language. Much as I hate citing the crass and shoddily written UrbanDictionary, their entry accurately defines that demographic's usage of "guilt trip":
When someone tries to make you feel guilty for thinking/feeling/doing things a certain way.
This is considered a manipulative, sometimes passive-aggressive conversation tactic.
Meanwhile, @FumbleFingers is right about the construction "X city", which you can also find on UrbanDictionary since it's the same level of language in terms of the generation that uses it and the contexts in which it's used. It means a situation where a lot of X takes place.
The character is very informally lamenting that if s/he says no to the party, s/he will be in "guilt trip city". We now see that s/he means: "People will seriously make me feel guilty for saying no."
The other character's reply then implies that it will be the character's dad who "guilt trips" him/her, that is, makes him/her feel guilty for saying no, and wants to know what he'll use to do that.
Here's an example of how the dad might "guilt trip" the character:
— "Dad... I don't think I want to go to the party."
— "Oh, so you want to break your mother's heart?"
Edit: Based on the other users' intuitions, I expect "guilt trip" did once mean what the Cambridge entry says: the experience of feeling guilty. But in this generation's parlance, it means the act of making someone feel guilty. The last character's use of "guilt trip" as a transitive verb shows that the video aligns with the newer use, which is the one I've encountered in my own 25-and-under peer group.