Two answers for you to consider.
1) From a standpoint of the NEC, you would either have to replace the breaker with a 15/20 tandem breaker or run 20 gauge wire as you describe. Some breakers are listed for double-tapped connections while most are not so look at the label. If they are crimped using a listed pressure connector that only presents one "wire" to the breaker then the connection is not double-tapped as far as the NEC is concerned.
15/20 tandem breakers are available, see
Square D by Schneider Electric Homeline 1-15 Amp 1-20 Amp Single-Pole Tandem Circuit Breaker-HOMT1520CP at The Home Depot
2) In practice no converter is going to draw over 15 amps (they have internal overcurrent protection well below that) and if that's the only connected load the hazard as a practical matter of 14 gauge wire is infinitesimal, particularly when you consider that with modern insulation, the ampacity of 14 gauge wire is much greater than 20 amps.