Whenever I plug in the game pad and start the game, the options start cycling without a way to control them.

Even in the setup program, after 'calibrating' the gamepad, in the test screen the pointer/cursor starts moving towards the upper left corner.

Any workarounds?

Try this - find the DOSBox configuration file (dosbox.conf - should be in Wacky Wheels\DOSBOX folder) and open it in notepad.
Change timed=true to timed=false. Save your changes.

Which one of all of the .conf files? There are several:

dosbox_wwheels.conf
dosbox_wwheels_client.conf
dosbox_wwheels_server.conf
dosbox_wwheels_settings.conf
dosbox_wwheels_single.conf

I have already tried this, following a tip that solved a similar problem with Terminal Velocity: I added these lines to the dosbox_wwheels_single.conf:

[joystick]
timed=false

It worked for Terminal Velocity, but it didn't work for Whacky Wheels.

As I said, find the one in DOSBOX folder and it should already contain the line "timed=true", just change it to false. If there is no such a file in DOSBOX folder, try opening any other .conf file from your list and search for "timed=true".

Party on, it worked for me. :D

You switched from "timed=true" to "timed=false", good, you're halfway trough!
Now, you must run the setup program (...\GOG Games\Wacky Wheels\Launch Settings):

Now do the following, once you opened the setup
1-) Select 'Controls'
2-) Select 'Player one' or 'Player two'
3-) Pick 'Joystick' or 'Game Pad/4 button stick' and recalibrate your controller
4-) Once you finished, you might want to test in 'Joystick/pad test' option

Hope I helped!

And for those who switched from "timed=true" to "timed=false" after calibrating the controller, you might want to recalibrate it, as the modification in the .conf file "resets" your controller device.

Happy Racing! (y)