Is there a gsm module that can directly send the peripheral data as a short message after receiving it?

No, GSM module is controlled by at command, but what kind of AT command is controlled by MCU program can't be controlled by GSM module. The GSM module is just like a mobile phone. Only when someone is operating can the mobile phone send text messages and make phone calls. Without the operation of the mobile phone, you won't send text messages or make phone calls. Operation is like at instruction controlled by single chip microcomputer program.