Wa alaikum al-salam wa Rahmatullahi wa barakatuh
If the fee that is paid to them is for the permission to play in the tournament, utilizing their courts, facilities, etc.., then it does not constitute gambling, even if the finalists are given prize money. The reason for that is that when one pays them the fee, there is a transfer of ownership that occurs with regards to the money that is paid. After this transfer of ownership, they are free to do with the money as they please, and giving some of that money to the finalists would be considered a voluntary gift (hiba) from them, which is permissible.
Allah knows best.
Answered by Shaykh Omar Mohsin