Although I have read about some men visiting the same escort twice in one day, the second booking is usually made after they enjoyed the first so much.

What your potential client is suggesting seems a little weird to me, and makes me wonder if he’s hedging his bets about which time he actually wants to see you. By making two bookings, he will either cancel or no-show for the one he does not want, probably on the day itself. As you take deposits for advance bookings anyway, absolutely take deposits for both although I suspect might not want to do this/will only pay one.

Personally, I would probably just decline as it all seems a little bit like hard work, particularly as you don’t think he’s genuine anyway.

Always listen to your gut, it is never wrong.

I have a regular who does this often when he's in town and likes to do an hour at lunchtime then another after work (I just charge him the usual two hour rate, atlhough it can be a pain having the lunchtime slot blocked out when I inevitably wind up turning somebody else away). He pays for both bookings at the start though, and I would make sure anybody doing likewise did too or I wouldn't hold the later time for them.

What the OP describes does sound dodgy to me so I would not be agreeing to that one.

The first time a new client booked me twice in one day was quite a shock and flattering. I went on to see him for several more bookings but not the twice in one day thing.  I have also had a younger guy (gorgeous  )  see me early evening and then come back mid morning next day, a few times. 

Another client I had met once only, and knew I didn't want to see him again, kept asking to book me at mid day then come back at 6pm but I had my excuses ready and he eventually gave up. 

I've already had someone coming back twice in the same day (both bookings were 30min ones) but he decided to do so after the first booking because he really enjoyed it.

But in the OP's case, I would find it very strange, especially if the client has never met you before and wants 2 bookings in the same day right away. The guy has no clue if you gonna get along well or not. In my opinion, it sounds like a time waster fantasist who won't even show up to the first agreed booking.

I've done this a few times but majority of the time they book the next day or within the same week. I recently had a reg who booked twice a week but after I asked him to wash (after he left a skid mark the last session) he was soft the entire time. He told me he will be busy for a few weeks but I don't think he is coming back now 

I’ve had punters come and see me once and then return next day or the day after but it’s normally for the 15 mins blow and go
I would ask for 2 deposits or get him in as a early morning booking so that if he fucks about its not messed up your day