The Proc SANHS article points to the full text of the original in Collinson's History of Somerset, volume 3, page 316, taken from the Bishop's register.
The foundation was by John de Bellocampo with consent of his mother Cecily, for 5 priests to pray for the souls of "domini Johannis de Bellocampo patris sui, Roberti de Bellocampo fratris sui, ... dictae Ceciiiae de Bellocampo, dictique domini Johannis, et dominae Johannae consortis suae, liberorumq; suorum Alienorae et Beatricis sororum ejus": that is his father John, his brother Robert, the said Cecily, the said John, his wife Joan and children, his sisters Eleanor and Beatrice.
On the next page there is a specification of money for distribution of bread to the poor for their souls, "viz. pro Johanne patre, Cecilia matre, & Johanne filio 20s. pro rata portione: & pro Roberto fratre, & Johanna uxore dicti domini Johannis dimidium marcae pro portione aequali": that is John the father, Cecily the mother, John the son, Robert the brother, and Joan wife of the said John.
The reference to 'John the son' I would take as referring to the founder himself in distinction from his father, rather than the son of the founder. The entry in the Regesta, from the Papal register, has therefore either misinterpreted this or was looking at a slightly different version.
The son John was therefore probably not born by 4 October 1304, or he would have been mentioned explicitly rather than under the catchall of 'and children'.
I agree with your assessment that the daughters are a phantom. CP ought to say "b. apparently after 4 Oct. 1304, when his father's sisters Alienor and Beatrice were living".