I'm told Kantians believe something like the following: that it would be inconsistent to respect our own preferences and not the preferences of others. If so, while pro-vegetarian arguments are often couched in terms of suffering and consequences, aren't there strong Kantian arguments for vegetarianism also? After all, many non-human animals do have preferences and desires, and generally prefer not to be eaten or killed.

Kantian ethics does appeal to notions of consistency, but the consistency that Kantian morality requires is not consistency in respecting "preferences" (as you expressed it.) Rather, Kantian morality requires consistency in respect for practically rational agency, i.e, the capacity to set one's ends and choose the means to one's ends. Kant believed (not incorrectly, for the most part) that non-human animals lack this capacity and hence are not owed respect in this sense. Indeed, Kant held that, strictly speaking, we have no moral obligations to animals at all; animals lack the property (practical rational agency) that lends something moral standing.

Many moral philosophers, including many Kantians, find Kant's conclusions troubling, inasmuch as it certainly seems intuitively plausible that we can wrong animals. Kantians have tried a number of solutions to rescue Kantian ethics from this position. Here are some articles in that vein:
https://philpapers.org/rec/CHOADK
https://philpapers.org/rec/DENKCO-2
https://philpapers.org/rec/TIMWTT
https://philpapers.org/rec/WOOKOD