You're a little confused
The FBO has ZERO requirement to show you anything, the renter has the OPTION of renting.
You come to my shop and demand to see the logs, I'll show you the door, you don't come into my place of business and demand anything. I don't have a requirement to do anything, including let you in my front door.
That being said, if you ask nicely and explain that you're flying your family, safety reasons, the children, whatever, I'm sure they will let you take a supervised look at the logs.
You should know this is, strangely or not, a odd request, most renters never ask, so don't be taken back when people at the FBO act slightly confused by your request. You would be well advised to make this request to the manager or CP ahead of time, as the logs might not be on site, or in a maintenance hangar with different hours than the FBO