there are pre-made SSRs in the voltage and current ranges you're talking about.... neat little 4-terminal blocks that can be mounted to a heat sink if need be... use an AC model for bidirectional use.
If you are switching DC, you can't use an AC relay, it must be a DC output unit such as this. The output is polarized so you have to place it in the circuit with the proper polarity. But it will work as either a high side or low side switch without modification since the input is opto isolated from the output.
actually, the MOSFET and bridge idea will work well. the bridge needs to be able to handle the current, and so does the fet. the gate drive needs to be referenced to the fet source, but isolated from the switched voltage, so that may be somewhat of a challenge. let me work out some of the details and i'll post a better schematic. what you see below is the conceptual schematic. R1 is the load. D1-4 is the bridge