If someone wanted to do both hunting and a little fishing in the mountains of the south island.

Is there any time of year that offers the best of both opportunities? Which species of fish? Which species of game?

Highlander will be able to answer that.

May/June????

Fish with be browns and rainbows. Mostly all the South Island rivers that flow out of the mountains into the lake systems close to fishing at the end of May.
I often hunt stags and fish in late March and anytime in April. Same can be said for Tahr.
The later the hunting the tougher the fishing gets, May you are more likely to find a duck hunter on the river than an angler and with less light and a lower sun angle its tough to spot fish.
Remember we hunt our fish ,especially backcountry fish, so you must have the sun.

March and April appears to be the answer.

I understand the sight fishing game. A bad wind will really make it difficult as well.

Do you primarily use spinning or fly tackle?

Pretty much 100% flyfishing. The vast majority of our waters at that clear that unless you are in very heavy water then the spin lures will scare more fish than its worth.
If your a very good fly fisherman then thats the way to go if in doubt grab the spinning gear.
Nice thing about march /April is that the winds are normally not an issue its right now during spring that we get our butts kicked by wind.

Early in the season, January , February and march you could hunt for Chamois for a few days then come down to the rivers for some Trout fishing.
This can be done along most of South Islands Alps.
Its not often you will find a hunting guide that is a guide for fly fishing so if you plan to use a guide "highlander" could be a good choice.