Ms Begum, 19, was one of three schoolgirls to leave Bethnal Green in east London to join the terror cult in 2015 and resurfaced at a Syrian refugee camp earlier this year.
Mr Javid cancelled the teenager's British citizenship in February in a move only permissible under international law if it does not leave the individual stateless.
"It was a unilateral, unprincipled response to the publishing of (Times reporter) Mr (Anthony) Loyd's interview (with Ms Begum in a refugee camp), deployed as an artifice or device to further your own personal political objective of being Prime Minister.
"Ms Begum was a pawn to your vanity. Her baby died."
Ms Begum, then heavily pregnant, was interviewed at the Al-Roj camp in northern Syria on February 13 this year and said she wished to return to the UK, as the self-styled caliphate collapsed.
Mr Akunjee wrote that she was concerned about the quality of healthcare which would be offered to her then unborn baby, given that her previous two children had both died.
The baby, named Jarrah, was born on February 16 and died of pneumonia on March 8, the letter said.
Stripping citizenship is only legal if the individual has a second one, and it was thought Ms Begum may have a claim in Bangladesh because of her family background, but Bangladeshi officials denied this.
The letter said: "Shamima Begum's parents never contemplated a life for her in Bangladesh. They did not register her birth with the Bangladeshi High Commission.
"They did not take her to Bangladesh on holiday as a child. Indeed she has never visited the country.
"Rather, Shamima was born, raised, groomed and radicalised here in the UK.
"The suggestion that Shamima is to you genuinely a Bangladeshi citizen is unsustainable."
It added: "It is the responsibility of a British Secretary of State to deal with British problems.
"Rather than take responsibility for Shamima Begum and her son, you took a British problem and illegally dumped it on our innocent international neighbours.
"You have dishonourably left it to the Kurdish people to bear the financial and security burden of Shamima's safety and upkeep.
"You left them to minister to her sick child and to bury him. Through sleight of hand, you have sought to burden the Bangladeshis with her in the longer term.
"Your cynical decision amounts to human fly-tipping."
The Home Office said it did not routinely comment in individual cases.