shower <<< primary tank <<<< supply tank <<<< cold water

You don't have a lot of tank-age, the inrush of cold water is chilling the supply tank. It's not recovering fast enough. The primary tank isn't triggering either until too late.

From the description, it takes both burners running to keep up with the demand of a shower.

To get both to trigger, each is going to have to see an inrush of what is relatively cold water. So the primary tank needs to be set to a higher temp than the supply tank. The tricky part is going to be figuring out how much lower the supply tank below the primary. Trial and error should work.

You may still see a drop from the initial temp, but hopefully it won't be such a fast drop as to be noticeable.