You are right, the other side of the converter is at RS232 levels, so it seems we are still lacking a TTL - CMOS level converter. My understanding is that connecting the Arduino pins to the JP6 pins bypasses the MAX232 and may damage the GSM module.
I went through the other documentation from the Sainsmart site but I found it less than informative, and I'm afraid that only people who dealt directly with this device can provide guidance. Here in the forum I found this Arduino Forum creative solution, where the poster created a loopback on the MAX232 and (apparently) managed to obtain the required conversion. There is no explanation, however, and the desired effect depends on having full access to the MAX232 pins and on powering it with a 3V source. I also wonder whether there's any undocumented accessory (such as the jack connectors shown in the photo) that can perform the trick.