Dear Norman
Regarding,
"... I tried using mail merge and I can get pretty close but I can't step over the finish line." ...
"... I need it centered because when I load the tray with the actual 3x5 index card, it's going to be in the center of the printer's paper path."
... "On one row I'd like to put the phone number (aligned from the left margin) and the hours of operation (aligned from the right margin)."
We can only fit 4 Index cards in a regular paper size (Letter or A4).
So we created a custom mail merge label template in the word document using an Excel Database.

In the sample picture below we have the cards for 4 museums in New York.
OBVIOUSLY, the final design is up to you.
I just wonder how much of your above statements match the resulting Index Cards template in the picture?

You may find in the following link a folder with the Word and Excel files with the above results
https://we.tl/t-L8pC2QCpFa
If you need more help
Kindly suggest preparing and uploading the Word and Excel files and sharing the link here using the instructions in the following video.
This will help us to give you a prompt and right solution to your problem
Regards
Jeovany
https://www.youtube.com/watch?v=NnXsE0SNuCc&t=14s