Wanting a live unaltered recording is good.
(Lots of approaches are good mind you.)
You want to make the grid and tempo map and all conform to the live performance after?
That's kind of a lot of work! That's why I'd ask what that is for after that? What do you need to have the matching ruler and grid for?
Unless you have something like video to sync to. If you are making a live (ie not clicktrack) recording, just record and ignore the grid. Put the project settings timebase settings on 'time' to keep them out of your way.
You can always edit things. It would be a live slip edit anyway. No need to make the grid match that for extra work.
The sample rate of the audio is your "grid" for live work.
If this is the other direction and you need a final product to be grid consistent. Maybe it's for DJs requesting that or whatever. You can alter things after the fact. Might be easier to record with a click to begin with though.
Anyway, figure out which direction this is. Only mess with the grids if you need them. Otherwise get them out of the way.