Once LTC gets segwit, all LN development will move there because who wants to develop under crippled mode (no segwit) when you have LTC, which is a BTC clone which actually scales and has the network effect of the original alt?
There is only one problem with your scenario - hardly anyone uses Litcoin. In the last 24 hours were only 6,798 litcoin transactions. see
https://bitinfocharts.com/Why on earth would all the LN development move to a coin nobody uses, and hence no fee profit can be made for it
because nobody uses it?
As things stand only ETh is a possible threat to BTc, because people are using it (there were about 87,000 transactions in the last 24 hours on ETH)
Indeed, this can be a minus factor for LiteCoin. However, if the SegWit can be successful then we should expect people to buy big volume of LiteCoin in the hope of making some profits. In other words, this can open up for a pump and dump situation. Anyway, let's hope that another altcoin can shine its own star alongside Ethereum and those in the top 10. I am thinking now of buying some LiteCoin.