One way of avoiding paying the storage fee is to go through passport control and enter non-Schengen area once you arrive in HEL and store your carry-on in self-operated lockers (maximum width is approximately 9 inches) at Via Lounge. You can also have a quick bite and shower while you're there. Then exit non-Schengen zone and re-enter Schengen zone by going though passport control and off you go to Helsinki. When you return, go through passport control again and proceed to Via Lounge. I've entered/exited passport control on the same day and it hasn't raised any eyebrows from border police.

And, if you're going through the trouble of doing this, just take the bus to town. You'll have an appreciation of how efficient public transportation is in Helsinki if you take the bus.