After a day's wait, I will venture some thoughts. Is it possible? I may be overly optimistic, but I am usually pretty confident that anything is possible with enough time, effort, and ingenuity.

Are you limited to what is shown in your screenshot? If I were trying to get a "calendar week" total over all tasks, I would expect to use something like a SUMIFS() function, but the SUMIFS() function is not going to like the way your data are arranged in this table. Depending on how the screenshot fits into the overall project, I might look elsewhere for the source data that the Gantt table/chart are pulling from, or take this table and separate it out into actual calendar weeks. There is no way for us to know how this fits into the overall project from just a screenshot.

A couple of possible approaches:

1) If there is a different source data table and it is arranged in a nice database format, then either a SUMIFS() or maybe even a pivot table, should be able to compute totals against calendar week
2) Come up with some formulas based on the screenshot table that will tell my final totaling formula that task 1 week 1 is its own week, task 2 week 1 is the same calendar week as task 1 week 2, and so on.

As I indicated, I am confident this is possible. Exactly how I would do it depends largely on exactly what else is part of this spreadsheet.