I have trawled the indexes. There can see no births with mother Charlotte BEAMES and son Charles. Perhaps they forgot to register his birth. Wonder if he may have been adopted and that is the reason George's death cert stated no issue with Charlotte although his issue from his first marriage were omitted. Who was the informant on George's death cert in 1924?
Searched in all states including Tas and SA and finally found this in Qld...mistranscribed. Thank goodness you had her 2nd given name. I hope this is your Eunice Bertha

1901/B2710
BRAY Venis Bertha Rebecca
Father Edward Thomas Mother Emily Louisa Beames
I checked the PROV Wills index to see if George POTTER who died in 1924 had a will but nothing listed.
I see Charlotte Elizabeth remarried. Wonder if Charles is mentioned as issue of former marriage.
POTTER Charlotte Elizth
ALEXANDER David
1930 Reg#2309
and her death
ALEXANDER Charlotte Elizabeth
Father Beams Daniel Mother Rebecca SEABROOK
At Portland 76 years 1949 Reg#17268
Another source to check Charles Beames birth.
Cheers
Cando