Sounds like this guy just found out about a cool new tool and decided to blog about it. BGP can be used on local networks but it's total overkill for a docker situation where all of your instances are likely in the same rack (often on the same machine!). Ifyou don't even have an AS from ARIN/RIPE there's no reason to even touch this (as you pointed out, it's all the protocol is designed to broadcast to the public internet -- e.g. 'hey I own this AS which has rights to this net-block, direct packets in this fashion please!' Jeez.
I have no idea what the CPU overhead of running this is but I'm sure it's not trivial, especially if the BGP daemon is tuned to retain any significant amount of the whole BGP table (RAM/swap issues galore I'd imagine). Granted the article is titled 'one way'.. which is empirically true, it's a Rube Goldberg way of going about networking.
(n.b., OSPF is what people use for "BGP" within your own intranet, even when you have tens of thousands of boxes. It's called "border gate protocol" for a reason..)
(Sorry, I don't use docker so I can't actually make a constructive comment telling you what the canonical/right way of doing it is, but I can assure you, this is not it.)