One big file would work and is the easiest and makes the most sense in my opinion. A nice feature is that you can name the scenes on the far right with a number and BPM (example: 86BPM, or 113 BPM) and when you trigger that scene, the tempo of the entire project jumps to that BPM at whatever global quantization you have set. So this could solve your issue of jumping to different BPMs. If you wanted more of a fading BPM change, like a speed up or slow down, you could map the master tempo to something on your APC and slide or move the tempo with that. Or potentially even automate a tempo change in a clip if you know exactly how you want the tempo to change.
Lot's of options to do similar activities is one of the reasons I love Ableton the most.
I would try to the naming of the scenes first, it's super easy and I think can get you going in the direction you want.
It looks like from what I read, Live Lite limits your tracks and scenes to 8, limits your send and returns and your inputs and outputs. Also you have the baseline audio and midi effects only. I would just keep messing with Live Lite and see if you really like the workflow and think it's the best choice for you going forward.