Why would you need to move your coins around? Electrum-LTC (just like Electrum) gives you full access to your private keys...
On the "Addresses" tab (View -> Show Addresses), simply right click on the address you need/want the private key for... and select "Private Key" from the menu... easy!
Or are you saying that you can move the LTC to Exodus to put their LTC on a new private key, so they don't risk their LTC when claiming Litecoin Cash? If so, don't bother with Exodus... simply create a new Electrum-LTC wallet, move all your coins to that AFTER the fork, then use the private keys from the original wallet to claim the LitecoinCash.
That way, you only need to move your LTC once... and as an added bonus, you don't need to use Exodus
On the "Addresses" tab (View -> Show Addresses), simply right click on the address you need/want the private key for... and select "Private Key" from the menu... easy!
Or are you saying that you can move the LTC to Exodus to put their LTC on a new private key, so they don't risk their LTC when claiming Litecoin Cash? If so, don't bother with Exodus... simply create a new Electrum-LTC wallet, move all your coins to that AFTER the fork, then use the private keys from the original wallet to claim the LitecoinCash.
That way, you only need to move your LTC once... and as an added bonus, you don't need to use Exodus