What your client is telling you is NOT correct, in the "Set a different price for this file" mode, if he paid 10 for your game, he will always be able to download files that cost 10 or less.
This mode has its drawbacks. For example, you cannot do discount sales or bundles, because the files do not work normally, it is not enough to have "bought" your project, but it works based on Tier.
For example, if at Christmas you make a discount and you want to sell your game for 5 and someone pays you only 5, they will NOT be able to download the files that normally cost 10.
If tomorrow you decide to release an update with a different price, for example 12, you will have a big problem, because again, people who paid 10 can only download files that cost 10 or less and you cannot simply pay the difference (2), they would have to pay 12 again and surely this is your client's fear.
Unfortunately, many developers have used this method without understanding how it works and this has led to many customers being affected.
The official recommendation is that if you have 2 versions of your game, one free and one paid, you should create 2 pages, one free and one paid, that way you can make offers or change the price without affecting the people who have bought your game.