Hi Mr. Bergel, I think  we might have spoken before. Thanks for watching the video!

You may have asked a question that no one knows the exact answer to... estimates for birches appear to be between 80-150 years, depending on the species. River birch seems to have the shortest lifespan, but I couldn't find specifics on weeping birch. Below are some links that illustrate the lifespan range attributed to birch trees- only the first is a strong source I'd have great confidence in.

As for the acephate/ imidicloprid question, based on my research- the imidicloprid is liekly to remain in the tree far longer. This is good for controlling the pest, bad for potential pollinator interaction. Both acephate and imidicloprid have potential risks to pollinators, but the precise risk is undefined and depends on pollinators interacting with pollen that is mostly windborne. So far, no one's funded a study to look at pesticide residue in birch pollen in the year following application of either of these pesticides, so we only know there is some level of risk either way. I appreciate that you are thinking about the big picture with your bronze birch borer management. I hope my response is helpful. Best wishes! 

 https://bigtree.cnre.vt.edu/lifespan.html

https://www.ontwf.org/wp-content/uploads/2016/07/PaperBirch.pdf

https://www.gardeningknowhow.com/ornamental/trees/birch/how-long-do-birch-trees-live.htm