Would a token that plans to take an additional percentage Lumens for transaction fees be possible to create on the existing Stellar network?
No, it's not possible. The maximum you can do is to control who is allowed to purchase your token with AUTHORIZATION REQUIRED flag.
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Thank you for your comment! So it would need to run on its own network entirely? Isn’t it possible to fork the code to create this? – Patagonia Mar 26 '18 at 11:57
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You don't have to fork anything in order to run your personal network. All you need is to specify another passphrase and arbitrary ports (you might want to run Mainnet and your private Network on the same servers, otherwise leave default ports). Once the network is started, you may change transaction fees. And all lumens on this network initially belong to you. See the configuration example here. – Orbit Lens Mar 26 '18 at 12:21