Ok, so it does appear I misunderstood... I definitely don't mind using viaBTC if it allows me to transfer those couple of BTCs for as little as 0.00163 BTC.
Is it just a matter of choosing the option to "pay only the required fee of 0.00001 BTC/kB" in bitcoin-core
You said the transaction size is 16.2kB, right?
To use the free ViaBTC acceleration service, they require that you pay a fee of at least 0.0001 BTC per kilobyte. I suggest adding in an extra 0.00000001 BTC per kilobyte just to avoid any rounding discrepancies.
In the "Transaction Fee section, choose "Custom", and "per kilobyte". Enter 0.00010001 BTC in the fee text box.
0.00010001 BTC per kilobyte X 16.2 kB = 0.001620162 BTC total fee calculated by your Bitcoin Core wallet.
There is no need to sign up if you want to use their free acceleration service. You just need to click the "x" in the top right corner of their public service announcement (or propaganda depending on which side of the SegWit vs. Emergent Consensus argument you prefer). Then enter the transaction_ID of your transaction and prove that you aren't an automated process by solving the Captcha puzzle.
The issue you'll run into is that they only allow a total of 100 free accelerations per hour. They seem to reset that 100 count at the top of every hour, but the service is popular so if you wait too long into the hour you'll get a message indicating "Submissions are beyond limit. Please try later". It is best to try within the first minute or so of the hour. Since your clock may not be perfectly in sync with theirs, you may want to start trying a minute or two before the top of the hour and try every 15 seconds or so until you either succeed or it is a few minutes into the new hour. If you are a few minutes into the new hour and haven't succeeded yet, then you'll probably need to wait an hour and try again. They don't seem to have a limit to the number of times you can try.
If you are successful, you'll get a message, "Acceleration Succeeded". Once you see that message you no longer need to try to submit the transaction ID. Your transaction will be confirmed in the next block that the ViaBTC pool solves. They will receive the 0.001620162 BTC fee that your wallet created when it created the transaction, so there is no need to pay them separately. They typically solve a few blocks per day, so you shouldn't need to wait more than 24 hours for a confirmation once you succeed in submitting it.
Once your transaction is confirmed, all those small outputs will have been combined into a new single output. Therefore, the next time you try to use those bitcoins, you will no longer have the excessively large fee (as compared to the rest of your bitcoins).
Be aware that by using this service, you are taking advantage of the generosity and charity of ViaBTC. This is a free service they provide to the community for when users
accidentally pay too low of a fee and want to get their transaction confirmed. By using the service
intentionally you are taking an acceleration slot that someone else could have used. ViaBTC is agreeing to include your transaction instead of a transaction that pays a higher fee-per-byte. As such, they are choosing to reduce their revenue to provide this service.
On the other hand, they are also providing this service to take advantage of those that don't understand how bitcoin works and as a result pay too low of a fee. By offering this service, they drive Bitcoin users to their website where they put up a message that encourages these less knowledgeable users to take a stand against the developers that support SegWit.
With that knowledge, you'll have to decide for yourself how you feel about using the acceleration service.