Even power Schottky diodes are not capable of switching hard on and off at 7MHz, they will work but be inefficient.
You could parallel lots of 1N4148s but you would need a lot and they need to be matched, not easy. Remember that as you parallel them their combined capacitance is also increased.
The modulation can't be applied to the supply in a DSB system, it has to 'unbalance' the mixer and applying it globally through the supply can't do that. I would be very cautious copying the schematic in post #1 too, it has some serious design errors. The 741 audio pre-amp should not be driving a capacitive load as it will almost certainly cause instability, there is no DC return for the mixer stage, no stabilization of the PA bias and no diode across the relay coil.
Brian.