Prime Minister Sheikh Sheikh Hasina has earned praise for opening up the frontier, but she cannot expect much international help in either campaign, diplomats and experts say.
And Bangladesh's warm welcome so far could easily sour if solutions are not found.
About 430,000 of the Muslim minority have streamed into Bangladesh since since August 25 when attacks by Rohingya militants in Rakhine state unleashed a blitz by the Buddhist-dominated nation's military.
The influx adds to about 300,000 Rohingya already in camps around the Bangladesh town of Cox's Bazar. There is not enough food, water or medicine to go around and the human excrement on roads around the camps only adds to UN fears that serious diseases could quickly break out.
The country has reacted with compassion to horrific tales of rapes and killings told by the refugees. Scores of trucks of aid donated by the public are arriving each day in Cox's Bazar. But it is not enough.
"Bangladesh can't deal with this crisis alone," said Champa Patel, head of the Asia Program at the Chatham House international affairs institute in London.
"It is densely populated, poor and already home to a historically displaced Rohingya community. While currently welcoming, this could change if the situation becomes protracted without any clear end in sight."
At the UN General Assembly last week, Hasina sought global efforts to solve the problems of the Rohingya, stateless as Myanmar refuses to give them citizenship and unloved there.
She called for the establishment of safe zones for the Rohingya. Myanmar has not responded.
- Zero influence -
Hasina also wants pressure on Myanmar to take back the group that has been in Rakhine for generations. But neither Bangladesh nor Western countries who have been shocked by events in Myanmar have any real clout over its de facto leader Aung San Suu Kyi or the generals attributed with effective power.
"The Myanmar army holds the key to resolve the crisis, in the immediate and short term," Ali Riaz, a professor of Illinois State University in the United States and an expert on Bangladesh-Myanmar relations, told AFP.
"An extremist Buddhist nationalist political force has thrived in past decades in Myanmar," Riaz said.
Bangladesh has "few effective instruments of statecraft at its disposal" to make Myanmar take back the refugees, according to Zachary Abuza, a professor at the National War College in Washington and specialist on Southeast Asian security.
No regional group has shown a desire to get involved either. There have been demonstrations in Indonesia and Malaysia for the Rohingya. Both are members with Myanmar of the Association of Southeast Asian Nations.
"Bangladesh could try to work with Malaysia and Indonesia to pressure Myanmar through ASEAN. But that is not going to work. The Rohingya issue is very divisive within ASEAN," said Abuza.
China and India are the keys to influencing Myanmar. Both have supported the Yangon government out of economic interests, said Patel at Chatham House and there is little Dhaka can do with either of the economic giants.
They "have investments in the country that they would not want to be undermined because of the current crisis."
The Myanmar generals are not listening to anyone anyway, according to Abuza.
"The Burmese military leadership has factored in the diplomatic costs of its actions. I think it is important to understand that they have been itching to do this for a long time," he said.
The Rohingya problem could however pose "a security threat to the entire region" if extremist groups lay roots in the camps of disaffected Muslims, warned Mubashar Hasan, a politics professor at North South University in Dhaka.
Bangladesh has struggled for years with homegrown Islamis allegedly linked to Islamic State and Al Qaeda.
Bangladesh's internal worries -- it fears radical Rohingya could add to Islamist troubles in the country -- can be seen in restrictions on movement placed on the Rohingya. Telecoms operators have been banned from giving them mobile phones.
Even silenced and stateless, the Rohingya problem will not go away.