If the Fed wanted to "invest" in Bitcoin, it would kill it by soaking up liquidity and creating a liquidity-squeeze. Because the amount in circulation is fixed, it wouldn't have to buy all of the Bitcoin -- just enough to kill any utility. This is a downside of an asset with a fixed quantity.
Once the community discovered the liquidity-squeeze spike, it would lead to a run-to-the-exit with a price crash. The short-squeeze spike and crash would spread across alt-coins as well due to rush-in-rush-out and leverage across alt-coins.
Once liquidity is killed, liquidity-squeezed, and a crash happens, the FED / SEC could make it illegal citing volatility and losses of citizens.
The Fed would need to do this discretely and never disclose their purchases. If it became public, the community would probably choose to fork.