Is it possible to send out transactions without sending them to the mempool?
it's possible to spot such transactions if they violate transaction forwarding rules (aka standardness rules) but not consensus rules. for example, a transaction greater than 100kB is not standard but still valid.
Little tiny 'gotchas' aren't going to save you from a determined investigator. They are trying to follow a trail of evidence so they can produce more evidence. What you need is a clean break, so that the investigator hits a dead end and has no productive leads they can follow.
Just pretend you sent that transaction a millisecond before it was mined.