It seems especially wild, because according to Wikipedia, The Order of the Dragon was modeled after The Order of Saint George, who famously slew a dragon. Did anybody at the time, or in subsequent years, comment that dragons are an odd or even inappropriate mascot for Christian crusaders?
Wikipedia tells me that the "dragon" in order of the dragon was somewhat like an ouroboros, but I'm not sure that clarifies anything.
Fun fact for lurkers, in case this question goes unanswered: In an ironic twist, The Order of the Dragon went back around to being associated with evil. One of the members was Vlad II, The Prince of Wallachia. He named himself Vlad II Dracul in honor of his membership in the order. His son was of course named Vlad, Son of Dracul, or, in his native Romanian, Vlad Dracula. Or, as you might know him, Vlad the Impaler.
The answer is simple: the dragon in the insignia does represent the devil, and the precedence of the red cross of St. George over the dragon represents the Order's purpose to defeat the devil.
Sigismund of Luxembourg founded the Order of the Dragon in 1408 to consolidate his faction's power during a difficult and years-long attempt to secure his thrones in Hungary, Croatia, Germany, and Bohemia.^1 Boulton agrees that factional consolidation explains the Order's creation.^2 According to Mirnik, Sigismund's creation of the Order of the Dragon amounted to “propaganda.” ^3
In the Order's founding statute, it sets out the group's intent “to crush the pernicious deeds of the [] perfidious Enemy, and of the followers of the ancient Dragon, and [] of the pagan knights, schismatics, and other nations of the Orthodox faith, and those envious of the Cross of Christ, and of our kingdoms….”^4 According to Štekar, “the symbolism of the order is evident: the victory of the Catholic Faith over evil, represented by the dragon.”^5
The context of the founding statute implies that the Ottoman Turks are the “perfidious enemy”/“the followers of the ancient dragon.” Additionally, the statute also calls out “schismatics,” which refers to the Hussites. Although the statute expressly denounces “Orthodox nations” as enemies of the order, there is no doubt that the Order at some points anointed Orthodox members, including the Serbian rulers of House Lazerević and House Branković, and the Wallachian rulers of House Draculeşti (Vlad II Dracul and Vlad III Dracula).^6
To zero in on the part of OP's question regarding whether contemporaries thought the dragon was an inappropriate mascot for Christians, there is no evidence that people thought it was inappropriate. Conversely, there is a ton of evidence that dragons were incredibly common in East-Central European heraldry. Mirnik notes that “the dragon itself appears on some older coats of arms of various noble families from both kingdoms [Hungary and Croatia], as well as the Principality of Siebenbürgen [Transylvania], sometimes even before the establishment of the Order.”^7
Further, the “dragon is normally (though not always) winged, bears a crown and is colored vert (sometimes or).... The real dragon of the order also had a cross gules on its back.”^8 To translate, Mirnik means that the insignia had a green or gold winged dragon wearing a crown and with a red cross on its back. The red cross is that of St. George.
“The dragon typically has the tail wound around its neck, which we ought to consider as the Order's authentic dragon, sometimes biting its tail. [G]enealogists and specialists in armoury are not sure whether the dragon wound around the shield is the genuine order, or just its reminiscence. There are many instances in which the dragon slid onto the shield itself. In any case, the families boasting the Order of the Dragon around or on their shield considered this as a very prestigious detail. In Hungarian, Transylvanian, as well as Croatian armoury, there are more than 30 families boasting the Order of the Dragon” on their family crests.^9
Even though the dragon was expressly and knowingly a symbol of the devil, princes and nobility in East-Central Europe were more than content to use it ubiquitously in their heraldry.
Citations:
^1 Mirnik, Ivan. The Order of the Dragon as Reflected in Hungarian and Croatian Heraldry. P. 3.
^2 Boulton, D'Arcy. The Knights of the Crown: The Monarchical Orders of Knighthood in Later Medieval Europe, 1325-1520. St. Martin's Press, 1987. P. 349.
^3 Mirnik, pp. 2-4.
^4 Boulton, p. 350.
^5 Štekar, Andrej. Knightly Decoration of the Order of the Dragon. Numizmatični Vestnik: Glasilo Numizmatičnega Društva Slovenije V Ljubljani. Ljubljana, 2001.
^6 Skoumal, Karel. List of Members of Order of Dragon.
^7 Mirnik, p. 6.
^8 Mirnik, p. 7.
^9 Ibid.