I think that with the proper mouthpiece and some practice, euphonium can certainly do walking bass -- and not just "the equivalent." I've done some of that. You should have no difficulty playing in the bass or even some of the contrabass tuba range -- down to the F an octave below the bass clef staff, and what more might you want for walking bass? The instrument can play lower, and so can you -- with practice and the right equipment. Another way to look at it is that the 4-valve euphonium plays in the range of the bass trombone -- which often plays bass lines.
You might want to take a look at Building Walking Bass Lines for some of the theory and technique of bass lines.
However, ... surely it's the upright bass that should be playing the bass line and the euphonium playing melody? Your kid might get a lot more enjoyment out of that as well. Or you could go in this direction, substituting your euphonium for the trombone -- although it might be a bit advanced to start with ... Elgar: Duet for Trombone and Double Bass.
Gary Merrill
Wessex EEb Bass tuba (DW 3XL or 2XL)
Mack Brass Compensating Euph (DE N106, Euph J, J9 euph)
Amati Oval Euph (DE 104, Euph J, J6 euph)
1924 Buescher 3-valve Eb tuba (with std US receiver), Kelly 25
Schiller American Heritage 7B clone bass trombone (DE LB K/K10/112/14 Lexan, Brass Ark MV50R)
1947 Olds "Standard" trombone (Olds #3)