You can ride your bike up the Rakaia ok, but the river crossing itself can be an issue. During April the river runs around 300 cumecs at the gorge and up further it's less, however a good Nor'west storm can bring it up to 1500 in no time. Same applies for the Mathias as you will have to access it by crossing the Rakaia at Manuka Point first. Wilberforce is a little different, the track up the true left is usually clear and only has small crossings until you get to the Unknown junction.
You look to have done a good bit of research, well done. Don't underestimate our back country as our narrow little country has some very diverse weather changes.
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