As above. Get your own Bitcoin wallet, and one that lets you and only you fully control private keys. Electrum as mentioned above is also the one I recommend - easy even for a newbie, with just a few minutes reading about how to set it up (otherwise, even just installing and going through it, ensuring you keep all information, should be enough to get you started). As you get more experienced and want to do more than just have a wallet, you can always choose to use another client.
That way, you never have any minimum amount or any other hidden restriction on limits (other than those set by the network). Keeping track of Payeer's $6 amount is going to be a bit ridiculous (especially if it doesn't tell you what that amount in Bitcoin is, and I'm assuming since you weren't aware, it's not apparent on the wallet).
So the $5.61 are now in Payeers possession but their costumer support says that they are not going to send the BTC to your wallet?
How you asked them if you can pay an additional $0.39 (to respect the $6 minimum) and will the BTC be sent to you in that case?
Or make a new $6 transaction and get $6+5.61 (from the 1st transaction) credited to your wallet?
Not sure this is good advice, they'd never bother. It doesn't make their job any easier anyway, if they were to do something about it. Only way to get your coins is to contact Payeer support, ask them how to resolve. Doubt they'll credit it to your account. $5.61 lesson learnt.