If he had done the exchange at the very begin of the manga, it would have helped him in a sense of the FBI stalker, but not towards killing L as he doesn't show up until late stages.

Light and L also cross-paths a few times however Light doesn't know his real name or face, just yet.

L also left to Watari that is anything were to happen to him, that the suspicious raised towards Light would be more than just probabilities.

So in short until Light is already caught on L's trap there was nothing that knowing his name would help with and by that time Light already had Misa by his side so there was no point to it.

>!Also keep in mind the Ryuk can see his lifespan and he probably already knew Light would die some time soon.