I wouldn’t bring any crampons or an ice axe as previously mentioned the chances of snow during the time period your planning is less than later in May / June.
Unless you have experience with crampons / ice axe then you’ll just get yourself into trouble.
As for waterproof trousers, get a pair of US ex army over trousers preferably with a side zip for easier donning & removing.
For a few years I just used a pair of ex British army over trousers I bought from an ex army store and they were perfect.
Gaiters, I use a pair of kuiu gaiters for our scrub down here, they’ve been solid, dependable and have with stood a tonne of abuse.
I would also recommend hiring a PLB (Personal Locator Beacon) when your down here, you can hire them through a variety of places, bivouac stores hire them out.
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