Hi Jessie45806!

As far as I can tell I think a Round Robin event should take care of your needs, but maybe I am misunderstanding your intent. I’ll break down my answer in two parts based on whether you want your client to meet one-on-one or with your team as a group. 

1. If you intend for your clients to meet one-on-one, based on who is available from your team, then a Round Robin does just that. It pools the three linked calendars from you team and shows all availability. Then, based on the time your invitee selects, assigns that meeting to someone on your team. As an admin you can even select who gets priority if multiple calendars have availability. If needed, you can always add other teammates to the meeting once booked.
2. If you want your client to meet with the entire team at once, then that would be a Group event. This event type also pools availability from your team but only shows open slots for when everyone has availability at the same time. 

I’m leaving a few resources below about Round Robins and Group events. If you have some other intent that I am not understanding please, come back and give us a shout. We want to help!

Round Robin events

Group events

You have the need correct. Pool our 3 calendars, show the client all times.

Well, then I am confused as to why the group availability does not show me available on the 3rd but my personal does. 

Yes, I have the same availability assigned to the events. 

That is confusing! It would be impossible to tell without me looking at your linked calendar and specific availability settings for each event type. Often there is some conflicting rule between specific availability, time zone settings, buffers, start time increments or an event on your linked calendar that conflicts with those rules. In this specific situation it might be faster for you to reach out to our Customer Support team as they can walk you through live and diagnose your issue in real time. Best of luck!