Do you happen to have a picture of this?
because I ordered this base in hopes of doing exactly what you describe, but I'm not sure if its going to work.
however, a picture of your setup would be invaluable.
I have had a lot of success using RAM branded mounts. I like them because I can swap out a cup holder for a phone holder, or a million other things. They are not the least expensive, but they are rock solid enough to be mounted on motorcycles, so you know it's not going to fail.
Here's an all-in-one solution:
RAM Mounts - Level Cup 16oz Drink Holder
If you prefer a holder for a taller coffee cup, the holder linked below would be more adequate. It would also require buying an appropriate 1-in ball mount and connecting arm.
RAM Mounts - Level Cup XL 32oz Drink Holder
I'm a huge fan of the company. I own 4 XL phone holders, a wheelchair mount, a suction cup mount, a clamp mount, a square mount, and several connection arms. None have ever failed for me.
This is the mount I purchased for mounting things onto my wheelchair's armrest frame: