Is there a single-word verb equivalent to "make something more professional"?
Solution 1:
It seems kind of obvious and you may have dismissed it, but why not use professionalize?
From Dictionary:
verb (used with object), pro·fes·sion·al·ized, pro·fes·sion·al·iz·ing. to give a professional character or status to.
Solution 2:
Consider elevate:
2 Raise to a more important or impressive level.
Lexico
As in:
We elevate scholastic chess.
For a phrase, consider take to the next level
Further improve or develop something that is already successful.
Lexico
As in:
We take scholastic chess to the next level.