Hello Brad,
I don't know how much time you have. I looked in my very good NZ road map. There are many small roads along the SH1. But it's too difficult to explain and for crossing the big Rakaia River you have to use the same bridge as the SH1. For this you need also a road map.
If you have more time, you can travel along the 72 via Darfield, Mt Somers and Geraldine through the Rakaia George. On this way you only have the last 25km to Timaru on the Highway. The way along the 72 is about 35 km longer than the the SH1 and in total with about 600m altitude.
If you want to continue to Oamaru and have more time, the cycling guide "pedallers paradise" suggest to go along the SH8 to Twizel, Omarama to Oamara. It also explains an alternative to avoid some kilometers with the tourist busses. But it's a much longer way. So may be that's not the best way.
You should buy the small book "Pedallers paradise" and a road map.
I want to tell you about a speciality for cyclist. Have you ever heard about the Central Otago Rail Trail? Take the train for a wonderful trip from Dunedin through the Taieri George to Pukerangi and follow the cycling way from Middlemarch via Ranfully, Omakau to Clyde in three days. The cycling way is an old rail trail and a great trip through Central Otago.
Hope fully I was able to help you.
Have a good trip. When are you going? We may be will be there in NZ, but we want to cycle this time from Auckland to Wellington.
Yours Renata alias Tangelo