Given the criteria you have set out, here's my thoughts. all else aside the Tahr ballot is your best option for a fly in bull tahr. the core blocks with highest returns will see you with really good chances. Having shot Tahr and chamois with both compound and longbow, The west coast scrub and broken country makes it easier to get close, but, Recovery can at times be difficult. What I would do is try get a ballot, and build in a back up plan as well if because if the weather is shit its not worth heading in even if you can. Also most North American and European hunters, except alaskan and coastal Canadian, do not realise how incredibly wet it can be.
Id try for a ballot, if successful work your timing to do it at the start of your trip so that you have some built in leeway if you can't fly out on time due to weather, If it all gos well and you still have a spare week, talk to James Scott at Karangarua about flying onto the tops for a chamois. You will need a partner for the ballot and anyway I don't suggest hunting any of that country alone.
Yep as others have said, it is doable to hunt with the bow. But also I have spent quite a few trips where I came home empty handed too. If you are happy to use a rifle, then explore that option and look at bringing your own rifle over. You have time to go through the process. Im on the coast and can supply you with storage etc if you need it.
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