Once you factor in a rear input for a small cab, you'd exceed your 13" dimension. Some small combos have their input, connections and controls on the top. If it's 11 or 12 inches tall, will there still be room for the input cable, and to make adjustments? I know you could probably make use of 90 degree connectors, and cut out or leave off the back of the shelving unit. I'm just thinking out loud here.
With size being determined for you, it may just be a reality that you have to go with what will fit, sacrificing tone and fullness. I'd say, unless you're willing to go with a good cab taking up the whole cube, you might not get much quality tone. You're also likely stuck with something that has all controls and inputs on the front being a requirement.
If you are willing to make your designated cube into a speaker, I have a build picture here of a cabinet that measures 12.5 x 11.5 x 12deep. You could change the input connector's location to the front. It costs around $100 to build one.
With size being determined for you, it may just be a reality that you have to go with what will fit, sacrificing tone and fullness. I'd say, unless you're willing to go with a good cab taking up the whole cube, you might not get much quality tone. You're also likely stuck with something that has all controls and inputs on the front being a requirement.
If you are willing to make your designated cube into a speaker, I have a build picture here of a cabinet that measures 12.5 x 11.5 x 12deep. You could change the input connector's location to the front. It costs around $100 to build one.