You can use "weekly withdraw" option to save some BTC.
If I recall correctly this weekly option costs "only" 3K satoshi.
As the name suggest it happens once per week, usually on Sunday afternoon, after the lottery draw.
I would say that you bitcoins are reasonably safe there ( if strong password etc) so you can easily store them there till upcoming Sunday.
Note: You will get small % of your deposit every day, till the day you withdraw. It was like 4% per year, I think.
If this is your first contact with BTC, use 2factor for your account and store them with freebitco.in. (always a bit risky though as you are not in full control)
I think he is asking how to transfer btc from bitbay to freebitcoin without incurred large withdrawal fee?
Yes, you're right, I totally misunderstood, not freebitcoin but rather bitbay fee is the case.
This is the reality OP, colossal fees... withdraw with altcoins seem OK but there is absolutely no way for you to save on BTC TX.
Everywhere you go it will be the same story, bitcoin is just damn expensive.
speedup-faucet.com's idea could do the trick, but what if no one buys your alts at faucethub? You will be waiting for days.
Private deal will be your only option, i think. Sell doge to somebody with small loss, maybe someone here will buy.
Or leave the bitcoin on exchange (risky) till you gather some more money for the tx fee.
Let this be your first lesson and hopefully the last. Read and think what you are doing. Sorry for misleading comment earlier.