I can only find one reference to Gollum's mode of speech in the Letters.
I believe there is only one error remaining in the text from which the Puffin was printed: like for likes (6th imp. p. 85 line 1; Puffin p. 76, line 23). This crept in in the 6th imp. I think. Not that Gollum would miss the chance of a sibilant!
The Letters of JRR Tolkien Letter 236 (to Raynor Unwin)
This doesn't tell us anything about what inspired that mode of speech. I think it is clear that this mode of speech is unique to Gollum and is not a "language" like Quenya or Finnish, or even a dialect like Northern English. I find tenuous support for this in the draft of a letter that discusses the customs of Hobbits.
There is no reason to suppose that the Stoors of Wilderland had developed a strictly 'matriarchal' system, properly so called. No trace of any such thing was to be found among the Stoor-element in the Eastfarthing and Buckland, though they maintained various differences of custom and law.
The Letters of JRR Tolkien Letter 214 (draft of letter to A. C. Nunn)
Tolkien points to the Stoors of the Shire as preserving the customs of their ancient relations in Wilderland. It seems likely that if Sméagol's people spoke like Gollum, that would be reflected in the speech of the inhabitants of the Eastfarthing and Buckland.
Sméagol/Gollum has two modes of speech. The corrupt Gollum character uses the distinctive mode with double pluruls and third person references to himself. On the few occasions when the more benign Sméagol character comes through, his speech is normal.
It seems clear to me that the Sméagol mode of speech is that of the people he grew up with and the Gollum voice developed over the years of Gollum's isolation and corruption.
So where did Gollum's mode of speech come from? As I said, Tolkien doesn't appear to have addressed this. However, I think a parallel can be drawn between Gollum's mode of speech and that of very young children. Very young speakers have not yet acquired the instictive understanding of grammar that older children and adults have; they make grammatical mistakes such as the double plurals referred to in the question. So my conclusion is that simply meant to convey the impression that Gollum is not sophisticated.
Of course we know that the character and speech of Gollum was originally developed before the Sméagol backstory. I have tried to give an in-universe explanation.