I have been able to get bridge in the correct place while the strings are in tune. After the locking nuts go on, moving the fine tuners causes the bridge to move up and down slightly.
No, I haven't used a block to lock the tailpiece in place while tuning. I guess I can give it a try, but I don't see how that's going to help since I've been able to get the bridge level.
Thank you for your responses and I welcome more.