The bar has good wifi. It looks like if I had my whole computer in there, I could cast the whole page that the video's playing on, but not the video alone, as you're right that there's no direct casting option.
Chrome on the ipad doesn't have a casting option at all, but I just found some ipad app that's a browser-ish thing and it ID's the videos on the page and makes them available to cast. Looks like it works alright on my home setup with the free KSTATEO videos that are on their page. Guess I'll take the whole setup to the bar when it's not too busy and try it out.
I know the TVs in the lounge areas of my building have the apple airplay/drop/whatever thing in or on them and it looks like that's available in chrome on ipad so that'd be an option too.