Any half decent engineer will be able to record you as live.
Don't dismiss everything that the engineer has to say but remember it's your money that's paying for the session so ultimately what you want goes (even if you're getting the session free, this still applies).
Many engineers like click tracks and recording the instruments seperately as they can take more control over the mix but if it isn't what you want, they should have no problem setting you up properly in the first place. It also takes longer and costs you more to record the instruments seperately!
And if recording live isn't working, you can always change tactics and go with the click track.
We record as live and then re-record the vocals seperately. This way we're using the vocals as a guide but I don't have to worry about them for the main take and can concentrate on my guitar. It also allows for a slightly different vocal take than when we play live.