I'd say you should take this route, rather than buying via local bitcoins (Your bank account will have a higher chance of getting frozen if you do)
How would your bank even know that you bought bitcoins?
I don't understand what you are saying.
Banks now have secret branches that report any sort of suspicious activity to the government. If the activity is suspicious enough they will freeze your account "pending internal review". Withdrawing any amount close to $10,000 would be considered suspicious. I'm just assuming from the title that "large amount of btc" constitutes around $10,000 but I could be wrong.
I was assuming that the OP already had the cash for whatever amount of bitcoin they were wanting.
Agreed, if the money is currently in a bank, and if the OP wants more than $9,000 worth, then they may trigger a CTR or a SAR upon withdrawing the cash depending on how they go about it.
Does Coinbase have a limit on how much bitcoin you can buy at once? You may find that the bank reports your transaction to FinCEN even if you don't withdraw the cash. Their risk assesment department might already be alerted on large transactions with companies such as Coinbase.