I'm only really familiar with lakes Okataina, Tarawera and Rotoiti (and Rotorua, but don't think that one would suit).
Okataina has no camp sites or showers, but plenty of bush walks and jigging opportunities. Given the lake being so low presently I'm sure there's plenty of exposed beaches where you could pitch a tent, but really no camping facilities so maybe not ideal.
Rotoiti ditto - the holiday park doesn't have camp sites available any more, to my knowledge. But the lake is great for fishing.
Tarawera is probably your best bet. Yes, you will need to book a tent site but that shouldn't bee too much of a problem. No hot showers, but toilets and hot pools to soak in.
You could also try Tikitapu / Blue Lake or Okareka Lake. The Top 10 camp ground is apparently very nice and it's just over the road from the lake. And I've heard the jigging can be very good. But I would not recommend them because they're only small and very popular with non-fishing boaties, so would think it'll be a nightmare trying to fish with jet skis wizzing past every few seconds.