Three different points.
A.) It is not the DBA's job to convince the CFO and that approach is ultimately doomed. The DBA should be working with the database's business side stakeholders (HR, Sales, Manufactoring, ....). Let them work the budget issues with the CFO. You may even find that the business no longer needs the database.
II.) is the approach to "moving" the tables and indexes. I inherited a database with all objects in a single file in primary. As the database approached 600GB we faced the physical limits of the then available disks.
After adding disks, file groups, and files; I moved selected tables and indexes using index maintenance tools. Non-clustered indexes were moved by rebuilding on new file groups and clustered indexes were moved with "Create - Drop Existing"
3.) The CTO will have to go back to school. You can't expect to perform major maintenance actions with "No performance impact". You can minimize and spread out the impact but you can't prevent it.
I have to disagree with the first point. HR, Sales, and Manufacturing have no clue in the areas we speak of. Their discussions may actually hurt the cause if they're not properly prepped for the discussion. If there's no CIO and no infrastructure group, then the DBA is going to have to be the one to produce and deliver the convincing argument.
On that third point, it sounds like the CTO doesn't actually need to go "back to school" for this problem and was directing an under-prepared DBA to come up with a better plan than just throwing hardware at stuff.
Sorry Jeff I disagree. It is the Business's responsibility to justify the expenditure. If they can't understand the value proposition then you should find a new employer. Sales says 'We have to have a complete record of every transaction forever". I tell sales that will cost you xMB/Transaction/Year. How many transactions do you want? That's pretty simple math. Protects me by providing a basis for sizing and makes them justify the cost to the CFO.
In that case, we'll have to agree to disagree.