The maximum current specified for a motor is mainly due to heat dissipation. That means that you can apply higher currents to the motor if it is only for short times, like with PWM ... Maybe (and only maybe) it could work. But it will be quite noisy ...
I understand you already have the L298 ... but I still think it is not suited for the scenario where you like to use it.
Maybe you could use some kind of PWM on the enable signals of the L298 board rather than in the own power lines ...