1. Are there better alternatives to Coinbase in terms of purchase fees and transaction?
IIRC you don't pay any fees if you buy them with GDAX instead of Coinbase. Go to Youtube and search "GDAX no fees"; There are tons of videos explaining it better.
2. Wouldnt it maybe be cheaper to buy LTC, move it to the exchange and buy BTC with it?
I don't think so. This would only be better if you wanted to buy Bitcoins to later send them to an multi-coins exchange and buy less popular altcoins like NEO, IOTA, XMR, other ERC20 tokens, etc... (due to LTC has cheaper miner fees and a fastar block time)