Ducted fans still run into the same issues if the airflow reaching the blades is supersonic. However with a ducted fan you can have the inlet slow the air down

Ducted fans still run into the same issues if the airflow reaching the blades is supersonic. However with a ducted fan you can have the inlet slow the air down to subsonic speeds. In this way ducted fans are used in nearly all supersonic aircraft today. It should be noted though that these turbofans for supersonic use are way smaller than those in use for subsonic aircraft. A 3 meter diameter fan is close to the current limit for such subsonic flight fans, scaling up a supersonic fan that large is still non-trivial. As for heating up the air and pushing it out, that concept is known as an airbreathing arcjet, which is really just a ramjet. While preventing the electrode from eroding is something of a technical challenge, it's comparatively mild. The main issue here is just power: batteries tend to discharge slowly so to get enough instantaneous power you need a very large number of batteries discharging in parallel, which are very heavy. However if you're just looking to set a speed record and not actually fly any useful distance, this may not be too much of an issue. Personally an arcjet would be my strategy of choice.