UPDATE: On 7/27, I submitted a ticket to Logitech support reporting that the link to the user guide for the K750 keyboard was dead and received their reply today. They provided to me this link for resolving the most common issues:
Scroll to "Changing the batteries in the K750 keyboard". Good luck!
Before reading the comments, I attacked this task just as stated by the author. My tool was a keychain-sized Swiss army knife. Now, I need to find a store that actually carries this battery. And as for making it easier to remove the battery the next time, I’m leaning heavily towards David Powers’ strategy; when I return the battery tray, I might leave the two pieces of the laminated business cards slightly protruding from the slot.