Radiant is magic damage, so you would have to run sun-fire (reflective light base skill) which does deal flame damage to be able to proc burning spell weave.

The tooltips say "deal [number] [type] damage" to indicate what type of damage an ability deals.

e.g. Sunfire states:

Blast an enemy with radiant heat, dealing 1161 flame damage, and an additional 3450 flame damage over 20 seconds.

Radiant destruction states:

Burn an enemy with a ray of holy fire, dealing 7242 magic damage over 3.8 seconds.

The parts I've bolded are the damage values and type of damage each ability deals. The non-bolded part before it is just the flavor text of the ability. So the "holy fire" of radiant destruction is just a way to describe the theme/visual of that ability, not the type of damage it deals.

So sunfire deals flame damage and radiant deals magic damage and since burning spell weave requires dealing flame damage to proc it, only the sunfire ability will proc that set since it's the only templar ability that deals flame damage.