Can I apply to international internship 3 month early?
I think you need to apply early as there is some delay in getting things set up properly before the internship begins. However, the question about whether you can start the internship early can only be answered by your current institution, which has made the rule. If they say it can't be started early, then the question becomes whether the employer wanting an intern is willing to wait for you to start and whether it will be sufficiently long for your needs. 

But you need to start asking those questions at the appropriate place (not here) immediately. Don't wait until December to start the process.