In theory -- yes.

In practice -- probably not. The problem is less the CIQ process or the additional ticket and more the total number of segments in one direction. You're at 5, which is one more than you can get onto a single automated bag tag. There are two ways for them to print your bag tag -- by hand, writing each segment down and eschewing any bar coding -- and by printing one tag to BKK and a second tag to CNX, which is then looped inside the BKK tag. You're going to find that a lot of agents (a) won't know how to do this, or (b) won't think it's possible. You can certainly ask, but I'd keep my expectations low.

I'd also make it a point to present my bag tag to each airline after UA as you get ready to board, so they can register it against your PNR in their system. And I'd put trackers in your bag if you don't already have them, although they're not all that useful when the bags are in containers. You can at least get an inkling of where the bag might be (but don't forget to check the timestamp).

If your bag does get misplaced, keep in mind that it's TG's responsibility to get it to you as the last-delivering carrier (whether you get the bags checked through to CNX or not, since they're operating your last two legs). This is true even if you know the bag is still sitting in BNA. They'd handle the communication with UA (or whomever) to get it to you.