Going nuclear is needed regardless of whether the Rats pull this nonsense. A better direct response would be to file a motion to Expel Kerry on the grounds that he's constitutionally barred from serving. See
this thread for details. To put teeth in the response, threaten to have the Senate subpoena his full military records including all the stuff he's refused to release. The information is needed to judge the case. As the Senate is the final word on its own membership I suspect the usual privacy restrictions could be bypassed, if the Congress hasn't already made an exception for itself.