Is it possible to upgrade my Time Capsule hard disk and keep the backup(s) from the original?
This is achievable for sure. 

First, you need to transfer files and then change your HDD.

This Apple tutorial shows how to transfer the Time Machine backups. 
Transfer Time Machine backups from one backup disk to another

Here in this iFixit link, it's a how-to tutorial for the HDD Replacement. 
Apple Time Capsule Model A1302 Hard Drive Replacement 

I hope you get it done!