I don't think so, maybe with a Keying group. Best workaround is to use an intermadiate object. Instead of constraining your character hand directly to the weapon, you parent an object, let's call it handCtrl, to the weapon and constraint your effector to it. When you want to adjust your hand position, use the handCtrl. This way you will not have to deal with constraint offset, just handCtrl animation, witch is more easy to manage.
Once this is done, you can use a character extension to key your handCtrl automaticaly in fullbody mode. To create the character extension, go to the Asset Browser panel, in the character foler and drag and drop a character extension to the handCtrl object.
Select the character extention in the navigator and go to the property panel. Set the attached character with your character. Verified that the Add To Full Body check box is checked.
Select your handCtrl, go to the property panel, click the Editor button. In the editor panel, go to Keying Groups, select Character Extention in the view drop down and add the local translation, rotation and scaling (if needed). So now you should be set and if you set a key in your character in full body mode, the local transform of your handCtrl should be keyed as well.
Let me know if something isn't clear.
Cheers
