I'm with you. 3rd March is a good pick for both what it represents (throwing off the shackles of the British) and for the time of year (early Autumn is basically still summer), and I strongly support the idea of changing it to that date.

I think I would prefer 3rd September or 9th October, the date the Statute of Westminster was enacted (depending on whether you go by the date it actually passed in 1942, or the date it was retroactively applied to in 1939). They both come at a time of year where there's a relative lack of public holidays. The 5 month period from late December to early May already have Christmas, Boxing Day, New Year's, Good Friday, Easter Monday, ANZAC Day, and Labour Day. That's 7. The entire rest of the year has just 2!

(Using Queensland dates here, but the trend is similar elsewhere, with a lot of state or local public holidays also being in that span. And, obviously, I'm discounting Australia Day from the 5-month period for the sake of the hypothetical. Otherwise it would be 8.)

Ideal would be an early November/late October holiday, to get that warm weather while still being reasonably removed from the Christmas–Labour Day period. The early October of the date the Statute of Westminster Adoption Act received royal assent is a little earlier than ideal, but has the added bonus of some synergy with ACT, NSW, and SA's Labour Day, as well as Qld's King's Birthday, giving two long weekends in a row: sometimes (including this year) a 3-day week, usually two 4-day weeks in a row.

But any of these dates would make great alternatives and have my strong support.