A tricky piece of the puzzle is that the liver, in addition to responding to blood sugar levels (like in a liver dump), it can also respond to blood *insulin* levels. In a non-diabetic person, since insulin levels also respond to blood sugar levels, low blood insulin can be a secondary sign of low BG. So sometimes the liver can respond to low blood insulin as if it were a low BG. I suspect given that this is a time of day when you've traditionally had lots of trouble with lows, that you have probably adjusted your insulin doses to be at their lowest around this time as well. So that may be a factor.
I'm not saying to take more insulin at the time of day when you've traditionally always been lowest and when you are doing exercise that's going to make it lower! Just pointing out that the situation may be even more complicated, in that you have multiple factors that all feed into a potential for a liver dump. Time of day (seems to be a point of high sensitivity for you), intensity of exercise, drop in BG, low blood insulin. With work and everything, you can't just change your workout time, and doing a lower-intensity workout may not be feasible for you either depending on your goals. And that's all perfectly fine!
But it does come together to mean that you may not be able to iron out that BG rise perfectly.
As a long-time insulin user, I have learned to accept that if I intend to do particularly strenuous exercise, a liver dump is always a possibility. There are things that make it more likely than others (working out before breakfast = guaranteed!) but even with the stars in alignment it's still possible. I found it easier to focus on the long-term benefits of exercise on my BG and my insulin requirements, or even to focus on the non-diabetic results of my exercise like physical endurance and strength, since the exercise itself actually played havoc with my BGs most of the time.
Of course you want to experiment, as you're doing, to find a routine that gets things the smoothest possible - find what works the best for you. Don't give up! Just don't be afraid to think outside of the box