How old is your daughter? She sounds very young. I haven't read your other questions, but need to ask whether or not your daughter is on the spectrum? Scripting like that is not atypical, but certainly unusual and very typical of children with ASD. When they don't know how to respond, they resort to a script.
My son was terrified of the bounce houses at 2 and 3. We also drove 30 minutes to one of those places. The place was HUGE, the party was not "private" and it was LOUD. For the first half hour, we also hung out where the shoes are stored. Eventually, he agreed to go on one of the bouncers is I went on with him. After a few rounds with me, he was off and running. Those places are a little overwhelming and I can see where she may resort to something comforting in that situation.
I would have done (and have done) exactly what you did. It IS draining, but I'm willing to bet that the next time you take her to a party... it's a little less overwhelming. Each time it will be a little easier for you both.
The ONLY thing you may want to consider is taking her for a walk outside if she resorts to her scripting again. That's her "tell" that she's overwhelmed (just a guess), so take her outside for a few minutes away from the noise and lights and let her do her kitten thing. Tell her that she has 5 minutes to play kitten and then she's going back inside.
Definitely not worth a meltdown, thought. It's draining, but she needs to see how PROUD you are that she went in for the pizza and cake, not how FRUSTRATED you are that she didn't go in right away.
ADDED: If it makes you feel ANY better, my son had a complete meltdown the first time we went to Chuck E Cheese because of the hand stamp. It freaked him out that they wanted to stamp his hand with "invisible ink". He was 3 and the 8 ft mouse was not an issue, but the stamp? Yikes. It was pretty awful, but what can you do?