You would have to pay back, you know that right? You have 20.000 ltc but you have to give those 50 btc back, ltc would have to go drastically up as you said but its impossible to predict that. Anyways 99% of loans here are for values worth < 1 BTC so they are obviously not for that
Why 'drastically'? Any up movement will give you profit (depends on the loan cost and tx fees). And why do you think you can't do this with <1 BTC loan?
Well basically as you said fees and loan interest would kill your profit. Like lets say 1 BTC = 100$ and 1 LTC = 1$
You have 120 LTC and you take a loan of 1 BTC to buy 100 ltc. you wait for the price to go up and you sell them lets say the price went to 1.2$ per LTC so you would have 120$ or 1.2 BTC (excluding any fees) so you basically won 20$ for that loan, now you have to payback the loan wich will have an interest of 5-10% maybe more in some cases. If it is 10% you have to pay back 1.1 BTC leaving you with 10$ profit (excluding fees) wich is a profit but too small considering all the things you have to do.
$10 is more than zero. Sure, the amount is small, but the whole investment is not big either. So you should focus on % rather than amount.
$10 is ~8.3% of your original $120 worth LTC holding. If we're talking about short term, say 1-2 weeks, that's very good result.
And if the period is short and the collateral is solid, you could easily find a loan cheaper than 10%. Those 'all the things you have to do' probably won't take more than 1 hour of your time combined.
Again, if you have X amount of LTC and absolutely no other funds available, and you know for sure that LTC will go up very soon, what do you do? Nothing?