Thanks for that. It doesn't look like dynamic pressure because that range of time has the slowest speed and altitude isn't that high. Your control loop is really trying to control two different rockets with one tuning: with the booster, and without the booster. I suspect that your controller is not as well tuned for the early stage with the booster than without. It may be helpful to look at the moment arms between the canards and the centers of mass for the two configurations. With the booster attached, the moment arm may be longer. This would essentially increase the proportional gain for that configuration.
You're right. The few second rise time during the 4-11 second time window seems reasonable. It doesn't seem too fast. However, whenever I see oscillations like this on a closed loop system, alarm bells go off in my head. The oscillation could mean that it is in a cusp of stability, and any disturbance could cause the system to go unstable. However, it sounds like you've already reduced the gain, so the oscillation may be due to "laziness" and not near-instability. It may help to add a sign to the tilt angle calculation. If the tuning is indeed lazy during this period, I would expect it to bounce around a mean of zero. Since your tilt angle calculation is always a positive angle, this can't be seen. Without adding a derivative feedback term (D), you may not be able to damp out this oscillation.