Well firstly, don't let that rego go until you absolutely have to! Rego might be expensive, but towing a car everywhere will cost far more (tow car + safe trailer + fuel/brakes/tyres when towing)
Many of the overly commercial/try hard events have "road registered" classes of some sort - but they meaningless because they don't require "road legal", just current rego. Pretty much anyone can keep plates on pretty much anything so there is no point sticking to a class like that.
The only other events to consider are rallies - these require registration of some sort to complete because at least some stages are on open roads. Full rego is by far the easiest, but most states except NSW have a rally/conditional/special interest rego that you can organise. If not, you can probably obtain a temporary "unregistered vehicle permit" or similar.