My 2 cents (likely overpriced) I was heading out today to purchase an HDMI adapter, and decided to check in here first. Best Buy has them in stock for the $49 as you mentioned. After reading these posts. Here is what I did to show my screen in a nursing home.
If the media you want to display is stored on your iPhone/iPad you can use an old router. No internet is required for the router to provide a network, which is all you need to use Air Play. It gets tricky if large files you are trying to display have been optimized into your iCloud storage.
Changing the wifi settings on the display and reconnecting to the preferred network can be avoided.
If the tv has an ethernet port use it. Saves connecting to your temporary network and then reconnecting to yours (or a hosts) regular network. Another way to bypass changing the wifi network settings is to use an AppleTV or Roku between the router and the tv. An HDMI cable connects the streaming device to the tv.