Thoughts about tri bar / trekking bar on hybrid bicycle
Aero bars (this is the actual name of what you call "tri bar") is surely a comfortable and affordable solution. In the Netherlands you see people with aero bars on hybrid bikes quite often - it is very common way to use the bike for a long commute and ease of manoeuvrability in the city traffic.

The only disadvantage of a simple version of aero bars is that you don't have the shifters and brakes there. For the shifters it's just inconvenience, for the brake handles it is also safety issue.

Bear in mind that aero bars have some pads designed for the elbows (or actually forearms) support and nothing (even thick wrapped bar) will beat it. 

Don't forget that the distance between the saddle and the bars should not be too long - when your body leans forward you shouldn't be stretching too much to reach for the handlebars. Otherwise you risk lower back pain on longer (in my experience >40km) journeys.

Concerning the bullhorn bars - in my opinion there won't be any added value for you except for the looks of the bike. And whether the brake handles and shifters are compatible depends only on the diameter of the bar you are replacing.