Tromp is a verb meaning to hike, step, stomp, or beat. It first appeared in 1892, apparently as a variant of tramp; from this sense we get the meanings of 'to walk around' and probably also the parallel to 'tramp down', i.e., to pack down or compress by stepping on. It is unclear if this is also the source of the meaning 'to give a physical beating to' and 'to defeat decisively', which may be influenced by stamp, stomp, and/or trump.
While tromp is a perfectly cromulent word, its comparatively recent origin and semantic commingling with many closely related slightly silly sounding words (-mp is an uncommon enough ending to sound odd) makes it sound informal and perhaps a bit child-like. Most writers and speakers will choose a stronger word unless they are trying for an informal tone.