Hi, Zabo!
In answer to your question: "Is there a way to keep the timeline integrity while adding transitions?", sorry, no, there isn't. This is common to many, or even all makes of video editing software. Each time you add a transition, you shorten the length of your video by the amount of time of the transition, e.g.: Transition time = 2 seconds; 5 transitions inserted, duration shortened by 10 seconds.
Best method might be to place all the clips(images) on the master video track,
then add your transitions,
then add your images onto the overlay(PiP) tracks, then add your audio. If the images are stills, compensate for loss of time by setting the stills times longer on the master video track, so that when you add your transitions, the amount of time lost comes back to the running time you initially invisaged.
Hope that's useful to you.
Cheers!
Neil