This description of bema based on how it was used for competitions makes it sound quite pleasant. Yet bema was used to describe the judgment seat of Pilate, who ordered Christ flogged, and then released Him to be crucified. Also used in Acts for when Paul is examined by the authorities.
I agree this is not judgment to condemnation, for Paul says:
1 Co 3:15. If any man's work shall be burned, he shall suffer loss: but he himself shall be saved; yet so as by fire.
This follows what he says in v. 13:
Every man's work shall be made manifest: for the day shall declare it, because it shall be revealed by fire; and the fire shall try every man's work of what sort it is.
In 2 Co 5:10 cited, Paul continues with:
Verse 11:Knowing therefore the terror of the Lord, we persuade men; but we are made manifest unto God; and I trust also are made manifest in your consciences.
Again, not the pleasant games award ceremony of the extra-Biblical use of the word bema that you cite.
It is for this reason that John exhorts us also:
1 John 3:3. And every man that hath this hope in him purifieth himself, even as he is pure.
2 John 1:8. Look to yourselves, that we lose not those things which we have wrought, but that we receive a full reward.
So we see that we can fall short of the full reward that has been ordained for us if we become seduced by the grace doctrine that suggests how we live does not amount to much because Christ already paid the price for our sins. We can even lose things that we have once wrought.
It is as you have well said, it is not judgment for sin, but it is judgment of faithfulness.
This is why Paul describes it as "he shall suffer loss".
Hebrews 2:3. How shall we escape, if we neglect so great salvation; which at the first began to be spoken by the Lord, and was confirmed unto us by them that heard him.
Thanks for bringing up this issue, as it can be a source of confusion, and a snare.
Peace of Christ with you and yours.