What would you do? Mine until you get to 0.02 and lose 0.01. Or call it quits, step away and know that they are keeping your BTC.
Yes, it was a surprise for me too when, on getting 0.001 btc (which is what bitcoin faucent gives away for free), I tried to send a payment and was told the minimum I can send is 0.01. It was another surprise when I realised, on reaching 0.01 that I needed to send at least 0.02 to make it worth my while - except I didn't see how 0.02 would make it worth my while as that would mean that I'd be spending half my btc on fees if I kept sending payments of 0.02 every time.
BTW, from the bitcoin wiki:
Transaction fees are voluntary on the part of the person making the bitcoin transaction, as the person attempting to make a transaction can include any fee or none at all in the transaction. On the other hand, nobody mining new bitcoins necessarily needs to accept the transactions and include them in the new block being created. The transaction fee is therefore an incentive on the part of the bitcoin user to make sure that a particular transaction will get included into the next block which is generated.
I'm assuming that, in this case, "the person making the bitcoin transaction" would be bcp, not its users (i.e. you and I).
And later on:
An advantage for bitcoin users to include a transaction fee is that the likelihood of getting a transaction included into the next block is going to be higher than if a transaction fee is not included. This is a trade off of time vs. money put forward on the transaction fees, as you can be patient with a low or non-existent fee included in a transaction, or you can make sure that the transaction is processed immediately by including a higher fee than is typical.
The rules are far from set in stone, and the network can support many different rules simultaneously. If there are 10 generating nodes that never require a transaction fee and your client is modified to never send any transaction fee, then your transactions will eventually be picked up by one of those free nodes when they generate a block, though it will probably take a very long time. In the far future, different rules about transaction fees among generating nodes will probably create a clear choice between fees and transaction speed. For example, you might choose to spend 2% for a guaranteed spot in the next block or 0.01% for the transaction to be sent in a few hours.
I agree with you that BCP should disclose their transaction fee up-front so that those using, or thinking about using, the service can make an informed decision. However, a little due diligence on the part of any prospective user of a service is always a good idea too. I decided to use it because I'm too lazy (as well as poor and miserly - I pay my own electricity bill and am not about to buy a massive fire hazard of a mining rig) to mine for myself but I do have a blog, which makes BCP seem that bit more appealing for me.
So, in answer to your "what would you do?" question: now that I'm already in it (whether having done the due diligence beforehand or not) I'd exercise patience and wait until I have at least a few multiples of 0.01 before I send a payment so that I'm not spending half my btc in fees on each transaction.