Hello Christian and welcome to our community!
Your asylum application(s) and Dublin law are not (directly) related to your wedding plans. If you and your girlfriend can provide the required documents then you can get married.
Both of you need to hand in a valid ID or passport at the Italian registry office along with a birth certificate, a certificate of no impediment and maybe some other documents too. Get in contact with the registry office where you want to get married to find out what documents exactly you need.
But you can also try to marry in Germany too, btw. That might be easier since both of you are already here.