No. The acoustic coupler modems from the old days were designed to "listen" to the sounds coming from the handset, interpret them, and then play back other sounds into the handset. This was a total bludge by today's standards, and even then they were only capable of 300 baud connections (far too slow to even try to access the Internet).
If your provider doesn't allow data transfer you are probably out of luck. Before cell phones had digital modems built in, they used to make cables that you could use to plug into the line-out of your analog modem so that you could use it as a phone line. Nowdays they don't make them like that anymore.