Maybe you can do a get list method when the user selects the time and the date

the filter would be greater than or equal to start and smaller than or equal to end date? put the result into a parametre, 

then do a rule if the parametre has a value, prohibit submission and show an error. 

you can also do a list view at the bottom of your screen to the same list, that would populate based on the picker controls in your form, once the day is selected the list populate the meetings/events for that day with the times and date, once the end date is selected, the view populates all the events matching the same start and end date.

would that work?

Zack, can you elaborate on "do a get list method when the user selects the time and date?" What exactly do you do here? I know you can "Execute a Method" in Rule Designer but for me there is no method called "Get List". The closest I have is "Get List Item by ID" and I'm not sure if it's the same thing. Everything in this method confuses me and I don't know what I'm supposed to do with it once I add it in Rule Designer.

Even if you were able to get a list of previously booked dates, how do you compare newly requested dates with previous dates if they span multiple days? For example, if someone booked Jan. 5-7 then a second person tries to book Jan. 6-8, how does the form know that Jan. 6 is unavailable? The only data it has to work with is start and end date and there's no data that says Jan. 6 is booked.