Conditional time format

Thank you, Sergei. That is beautiful. Unfortunately, given how I've had to manipulate my data to get the time to show "a.m." rather than "AM," I can't get it to work.

The initial problem is that I haven't found a way to get an actual custom time format to do "h:mm a.m./p.m." No matter what I've tried, I can't get it to put the periods in. So to accomplish that, what I've got is a hidden column (I) that calculates AM/PM and displays that as "a.m." or "p.m."
=IF(MOD(E3,1)>=0.5,"p.m.","a.m.")

And I've got another hidden column (F) that subtracts 12 from afternoon times so I end up with h:mm but not as 24-hour time:
=IF(MOD(E3,1)>=0.5,TEXT(E3-12/24, "h:mm"),TEXT(E3, "h:mm"))

Finally, I have a column that concatenates those together along with the date in the format I need:

=CONCATENATE(TEXT(D3,"mm/dd/yy"), " ", (TEXT(F3,"h:mm")), " ", I3)

There is probably a much more elegant solution, but that's the best I was able to come up with using my limited set of skills. I can't seem to find a way, either by conditional formatting or adding another link in my chain of calculations that uses the formula you came up with to get me where I'm going. I can get it to return 4PM as 16 or 4:15PM and 16:15, but I can't get it to show as 4 p.m. and 4:15 p.m.

Do you have any suggestions to get me to my desired and result?

Thanks so much!