Is it possible to beat this game without using Diddy Kong?
Obviously there are places where you will pick up Diddy Kong such as some mine cart levels and etc. What my question really asks is, if it is possible to only have a solo character run through the entire game. This would basically mean that you get rid of your second character as soon as you can in a level if you pick up one, and continue on without him.
I believe this would be especially challenging and maybe impossible considering some of the jumps that have to be made which are made easier with the assistance of a second character.
It's most definitely possible, simply based on the existence of this game's "Mirror Mode". After completing every level in the game, that mode is unlocked. In addition to everything being mirrored (right<->left, meaning you're moving left the whole time) and having only one heart (essentially daredevil mode), you also can't ever use Diddy. Having just finished beating every level in this mode, I can personally verify that yes, you can beat every level with just Donkey Kong.
Of course, if you would like to do so in normal mode, then yeah, you can just do what you said - avoid DK barrels, and just intentionally run into enemies to lose Diddy if you ever accidentally get him. If you want the challenge, it's certainly harder, since it limits your margin for error when making jumps and whatnot.
"Why is it so hard? ... Why can't you just take me? ... I don't have much to go... before... I fade completely..." - Kelly Clarkson