3.7V to 7.5V step up circuit
1.5 amps needn't be the limit when using the MC34063A chip: -

But, even if the peak current is limited why not run in CCM mode? If your input voltage needs to be as low as 3 volts (for example) and you used a 15 μH inductor, I can't see the peak inductor current exceeding 1.75 amps: -

The above calculation is based on producing 8.1 volts (7.5 volts after a non-ideal rectifier) at the output and driving a 20 ohm load from a 3 volt input supply. \$I_{PEAK}\$ is 1.723 amps. Switching frequency is 100 kHz.
I've also assumed that once the output capacitor is charged up to the required output voltage that it will &quot;soak up&quot; the 350 mA pulses to a certain extent. In that respect, I've made the assumption that the output current will average at 375 mA. The MC34063A has a current limit facility so I don't really see it having a problem.
Calculator can be found here.
If you used a 22 μH inductor, the peak inductor current is 1.523 amps.
If you use a 33 μH inductor, the peak inductor current is 1.380 amps: -