Our previous unit had the same AC as yours (13.5), and we did use it on 15 amp service sometimes, but can't run anything else. Fridge and water heater on propane, no TV or any appliances.
You can damage the compressor when it comes on if it doesn't have enough power. It will eventually trip the CB on the post but may be too late.
Keep an eye on it and if you hear it struggle to start the compressor, switch it off. I wouldn't leave it unattended. Some units have a built in safegaurd to drop the compressor out if the power is too low, check your unit.