Plugin tabs do not typically allow for fields. The activity tab is no exception here. However, plugin tabs usually render the tab description on profile view instead of profile edit and CB Activity is one of those plugins so you should be able to put your HTML in the tab description. Tab descriptions don't support content plugins though so you won't be able to insert a module there.

There's a few alternative options here though. You could move the Activity tab to a not shown on profile tab position. Next create a new tab in CB > Tab Management. Now add a Custom HTML field with the following to that new tab.

[cb:usertab tab="ACTIVITY_TAB_ID" /]

This effectively moves the display of the Activity tab into a Custom HTML field so you can add more content to it or add fields before/after it as desired.

Another alternative is to replace the Activity tab with an activity field.