It is shady.
Invoices are your business records. They reflect your income. You should never add anything to any invoice which your business is not directly responsible for.
The client is asking you to put yourself in an untenable position. Refuse to add random items or provide anything editable. No one should ever need to edit an invoice you send them.
It sounds to me like this client wants to pass on costs to someone else.. and wants to make it appear you are charging more than you did in an effort to hide his/her fee. Or, in order to better the client's position financially (taxes) they want to make it appear as though they paid you more than they actually did. It's absolutely shady, if not fraud.
If the tax man comes calling and wants to see your business records... and there's an invoice for an additional $x amount you added merely because the client asked you to, but never collected any actual money.. well.. the tax man will see that as income for you/your business and you had better pay taxes on this "invisible" amount.
I would explain to the client:
Invoices are business records. I will not allow my business to be liable for line items you request be added unless my business is directly responsible for those line items. In addition, I will not jeopardize my tax position with invoices which reflect inaccurate income amounts. What you are asking is unethical.
If that angers or upsets the client, for me, I wouldn't care. There's obviously some nefarious business practices taking place and he/she is trying to pull me into that and I want no part of it.
