Interestingly, OED defines travested as an obsolete adjective that functions like the past participle "travestied" and antedates all attestations of "travesty" as either a noun, adjective, or verb.
Obs. As past participle: disguised; travestied.
1656 T. Blount Glossographia Travested, disguised or shifted in apparel; And metaphorically it may be applyed to any thing that is translated out of one language into another.
Travesty as a noun or adjective is not attested until c1662.
c1662 W. Davenant Play-house to Let i. i What think you Of Romances travesti..Burlesque and Travesti? These are hard words, And may be French, but not Law-French.
1664 Cotton (title) Scarronides: or, Virgile Travestie. A Mock-Poem. Being the First Book of Virgils Æneis in English, Burlésque.
Travesty as a verb is attested third, and appears earliest in the past participle, similar to "travested."
1673 Bp. S. Ward Apol. Myst. Gospel 42 Are the Mysteries of this Gospel..to be travestied or turned into Burlesque or Macaronique?
1756 J. Warton Ess. on Pope I. ii. 57 One would imagine that John Dennis..had been here attempting to travesty this description of the restoration of Eurydice to life.
So in OED, "travested" as a past participle appears before "travesty" appears at all. The full etymology provided suggests that the word was, in fact, derived from a French verb.
Originally < French travesti, feminine travestie , past participle of (se ) travestir (Montaigne a1592), ‘to disguise him, or take on another man's habit’ (Cotgrave), < Italian travestire to disguise (Florio), < tra- = trans- prefix + Italian vestire, Latin vestīre to clothe.
This description doesn't address the continued popularity of "travesty" as a verb, but I thought it was interesting (and relevant) that the word is derived from a verb and appeared first as a past participle, which runs quite contrary to what we're used to today.