Their site says that it needs to be at least 0.0001BTC/kb which is around 10000 satoshi (If I remember rightly).
Yeah you're right here, at least 10k satoshi for the fee
Fee per byte which is sat/byte doesn't involve about it, the important is the total fee of the transaction which is he have just 0.0000565 BTC or around 5k sat.
No, you're wrong. You're reading the requirement incorrectly... it is
NOT based on total fee... fees are ALWAYS (and should always be) calculated using
fee per byte. This is why the fee estimators ALWAYS use BTC/kB or sat/byte because that is the value that miners use to prioritise transactions.
It says on ViaBTC:
We only provide accelerator services for those with at least 0.0001BTC/KB transaction fees;
That is 0.0001 BTC
per kiloByte or 10,000 sats
per kiloByte or 10 satoshis
per byte... His transaction was 25 sats/byte and as far as I could tell, met ALL the requirements of ViaBTC.