I am sure the introduction of exercise will be having a significant impact and quite possibly why your overnight levels are dropping. Exercise certainly does that to me and I have to reduce my evening basal dose to prevent nocturnal hypos after I have done plenty of exercise or been particularly active all day. Unfortunately Toujeo doesn't allow that sort of adjustment so you may need to ensure your levels are higher at night although it seems your body may be doing this anyway at the moment if you are rising into double figures in the evening, so probably best not to be tempted to stick a correction in at bedtime if levels are high at this stage.... assuming you have been told about corrections? .
And yes, it may well be that your own beta cells are reviving a bit during this honeymoon period, so that could well be a factor as I mentioned. You really just have to manage it as well as you can until your body gets into a routine and you can see clear patterns. It may be that a change of basal insulin might be helpful with a split dose Levemir enabling you to adjust your day and night time doses independently to allow for exercise etc or optimizing your Toujeo dose to give you a near horizontal line overnight and then adjust your ratios for during the day to balance it out.
I love Levemir for it's shorter action and the flexibility it gives you to adjust to what your body needs between day and night but some people love Tresiba and Toujeo which are the very long acting basal insulins for the lack of adjustment needed. I think it is good to wait until things settle a bit and you get your feet under the table and more confident/experienced before switching insulins because it can really disrupt your whole diabetes management and in the early days that can be unsettling.
So in response to your original question, yes certainly experiment with injecting your breakfast bolus after you start eating or just after you finish for a few days and see how that works out. Out of curiosity, does your Libre graph actually show you going into the red during breakfast or has it corrected itself and that dip into the red has mysteriously vanished from your graph as often happens.
