For what it's worth, I contacted the support team at Avatron (the developer of Air Display) and asked whether they had any workarounds. Here's what they came up with:
Thanks for contacting us at Avatron support about the issues that you've been seeing when connecting via an ad hoc network. I was unaware of this problem but after doing some testing this morning it seems to happen with one of our iOS 7 iPad 2s as well.
While this definitely seems like something that Apple needs to address, I was able to get Air Display working over the ad-hoc network if I disabled internet sharing from the Mac System Preferences > Sharing pane. Obviously the drawback of this is that the iPad will no longer be connected to the internet when you are using Air Display in this manner.
Please let me know if this is able to serve as a temporary workaround while Apple investigates the problem.
Cheers,
Aaron
Technical Support
Avatron Software, Inc.
Clearly this is not a full fix as I can't access the internet when connected to my ad-hoc network, but at least I got Air Display running.
Their reply also suggests that this isn't a blanket issue covering all devices running iOS7, as only "one of [their] iOS 7 iPad 2s" was having the same problem.