Your source code is your name. Having an additional name would be irrelevant. It is certainly possible for bots to prove they cooperate with a given bot, by looking at that particular bot's source. It would, as you say, be much harder for a bot to prove it cooperates with every bot equivalent to a given bot (in the sense of making the same cooperate/defect decisions vs. every opponent).

Rice's theorem may not be as much of an obstruction as you seem to indicate. For example, Rice's theorem doesn't prohibit a bot which proves that it defects against all defectbots, and cooperates with all cooperatebots. Indeed, you can construct an example of such a bot. (Rice's theorem would, however, prevent constructing a bot which cooperates with cooperatebots and defects against everyone else.)