Also who is the most reliable exchange or, converter?
n number of altcoin trading site available in online. You can use anyone among that sell your bitcoins to get the LTC in your wallet. I will suggest you to have the trade in bitrex, polenix or livecoin.net to trade your altcoins for bitcoins.
I understand you are worry about fees. That you have bear it from your side. Recently BTC transaction fees increased in high volume and compare in fiat cash also it increased big so that will be surely as your loss. Try to open a trade for high price. That is only way to adjust it.