Highly unlikely because the character name clean up is automated and only effects accounts which meet a certain criteria. From what I understand banned or frozen accounts are considered inactive but still there for some reason. I think you could possibly try to email them and ask if they would delete the other account. It's also possible there's some reason you're not allowed to take the display name someone used when they got banned.
Inactive usersEdit If a user is inactive (has not been active in a while), their name will eventually become available as a display name if the following conditions are met:
It hasn’t been logged into in the last year It hasn’t been used to purchase any membership It has no RuneScape skills of level 30 or above Note: If you are a free player, you will retain your character name if you have logged into your account in the past year, or if you have any skills over level 30
This is to free up names from inactive accounts for active users to be able to use for their account. Any players who choose to log into an inactive account that has had its name taken will be forced to change the name of the account before logging in. Display Name Interface The interface shown when one attempts to log in to an Inactive Account. In the summer of 2014 Jagex announced that more character names will be freed up in batches of accounts that follow the following criteria:
It has never had membership and it hasn't been logged into in the last 6 months It is a former member and it hasn't been logged into in the last 2 years The first batch of names beginning with the letters A, B and C was released on 5 August.