Is a solid state relay with over 1 MHz driving frequency physically possible?
So is it even possible for an SSR to switch this fast or should I have went with a different topology to deliver the 1000 V?

The design of solid state relays often prevent this. In order to not consume much energy when off they disable the booster required for the high side switching.
You can do this yourself with a high side gate driver and N-channel mosfet. Eg: 1EDN7550.