Even if you cannot legally hunt with a bow, IMHO, being skilled with a bow and arrow is far too important of a survival skill from many aspects to not learn this. It's also fun.
I've always preferred a more traditional style of bow, long bow or recurve, due to the fact that you can use a wooden arrow, unlike a modern pully style compound bow. You can make then, and personally, I like the way they fly. There are several styles of primitive arrow manufacture you can practice using both natural and scavenged materials. So, for the primitive craftsman looking for new frontiers to explore, manufacturing arrows and bow strings presents a whole new basket of opportunities.
I'm a certified firearms and an archery instructor in Texas where deer hunting is a serious affair, and bow hunting gives an extra season. One thing I can say as an archery instructor is that it's a skill that you need to learn far in advance of when it's needed. Getting good with a bow takes a while. You do this simply by repetition and lots of target practice, both stationary and moving.
If you are a primitive skills craftsman, focus initially on making the arrows and the strings, not the bow. Making a bow that isn't just a temporary, disposable quick fix is a serious project and requires a lot of skill and experience to get right. You will most likely screw up the first few you make. In fact, one of the oldest bow samples ever recovered was found in the UK in a bog and it was tossed into the bog because the maker screwed it up and it broke. Otzi, the Iceman, if I recall, was still in the process of making a new bow when he died. It's not something that even those with skill and experience do quickly.
Amongst the stone age tribes of North America, one of the budding cottage industries amongst some tribes was making bows which they would sell to other tribes. An example of this were the Yakima tribe and their lightweight laminated bows that used sinew and glue made from human spit and fish bladders for the lamination. These bows were in high demand from other tribes.
So, what I'd recommend is to get a bow that is premade, either a long bow or a recurve, and then practice making both the arrows and the strings using primitive skills. A good bow made with modern materials can last a long time, but the strings get worn and the arrows get used up, lost, broken, etc. These, however, you can make a lot easier than making the bow itself.
I'd look seriously at a take down bow simply due to ease of stowage and transport. Unless you are fighting a battle against armored knights, you don't need an incredibly powerful bow to get the job done. Arrows kill by hemmoraging anyway, and you need to be able to comfortably shoot it to get good with it.
Some neat bows are the folding bows from Primal Archery. They quickly fold out for stringing, or fold up to a compact 23 inches (58cm) long. The staves are replaceable for either repair or to change the strength of the pull.
http://www.goprimalnow.com/
For those that don't already know this, you should never fire a wooden arrow out of a pully style compound bow as the arrow can sometimes shatter upon string release. This can result in horrific injuries, specifically to your hand holding the bow. A friend of mine, who knew better, did this repeatedly and after a while he wound up with a shattered wooden arrow shaft that had gone through his wrist with long wooden shards of shattered arrow shaft sticking out between various fingers on that hand. He described it as the single most physically painful thing he had ever experienced, and the recovery with modern medical care took a long time.
