Not really, if you'd built your model from a billion launches and 20 million failures, a dozen failures in a row won't change your priors much at all. You've built your model off of far less data.
How many failures are required to change your model from 2% to, say, 4%?
It is easy enough to modify the linked script for any scenario that you want.
If 2 engines failed on the next launch, the model would predict a 4.8% chance of failure on the following launch.
As for your "build your model from a billion launches and 20 million failures" comment, if I had that much data, then it wouldn't much matter what reasonable set of priors that I started with, I'd wind up convinced that the true failure rate was very close to 2%.
Note that the prior that I am talking about is the distribution of possible theories before I saw ANY data.
How many failures are required to change your model from 2% to, say, 4%?