The following are two examples of Indian politicians threatening Bangladesh with invasion:
Has India ever threatened Bangladesh 'officially' with the military invasion?
Officially, no. Minor skirmishes were there at the border sometimes.
Further reading:
https://en.wikipedia.org/wiki/2001_Bangladesh%E2%80%93India_border_clashes
"There are enduring linguistic, cultural, and religious similarities between the people of Bangladesh and India that transcend their respective national identities. India played a significant role in Bangladesh’s war of liberation. India’s contribution to supporting Bangladesh’s liberation war initiated friendly relations between the two states."
Pro-India Phase up to 1975: The Pro-India Phase beginning from 1972-1975 was a great season for Bangladesh-India Relations to endorse friendly, peaceful, and harmonical cooperation and friendship. This season started as already mentioned due to India's support for East Pakistan (today's Bangladesh). Even the Pro-Indian leader Bangladeshi Prime minister Mujibur Rahman visited India, by which Indian prime minister Indira Gandhi followed, symbolizes the Bangladesh-Indian effort to raise cooperation. In addition to that, the signing of the 25-year Treaty of friendship in March 1972 consistently promoted more negotiations between the two parties, stable borders, strengthening of peace and security, and "settlement of disputes" to accomplish the UN Charta. "The Treaty made provisions for effective steps towards securing peace and stability in South Asia." During Mujibur Rahmen's reigning, cooperation was entitled in "fields of trade, commerce, education, cultural activities, and sports" (citeseerx.ist.psu.edu, n.d.).
The weakening of Bangladesh-India Relations: The Relationship between Bangladesh and India worsened over time as disputes began to rose since Anti-India propaganda flew across Bangladesh and as Indian concerns came across territorial disputes and illegal immigration from Bangladesh but also the assassination of the pro-Indian leader Mujibur Rahman of Bangladesh and also Chinese rising influence...
The friendly relations between Bangladesh and India did not last for long. Cordial relations of the early days shifted towards hostility and mutual suspicion. Antagonism between the two states emerged out of the anti-India propaganda propounded by certain vested interest groups and the Bangla press, considered to be a major negative development of the Mujib era. Apart from this, there were various other factors responsible for the anti-India propaganda in Bangladesh at that time. A few among the reasons included the existence of pro-Pakistan groups in Bangladesh, internal opposition to Sheikh Mujibur’s policies, Pakistani and Chinese propaganda about India’s imperialistic designs over Bangladesh, rise of communalism in Bangladesh, strong dissatisfaction within Bangladesh over increased corruption, shortage of essential commodities, particularly foodstuff and uncontrolled smuggling on the Indo-Bangladesh border
With the assassination of Mujibur Rahman, the shift towards hostility in relations between India and the new government in Bangladesh became irreversible because no other leader who was as pro-India as Mujibur Rahman emerged in Bangladesh (Franda 1982: 130). This incident also ended India’s hopes of restoring friendship with Bangladesh and the hope that Bangladesh would remain democratic.
A major conflict during this period was related to the border between Bangladesh and India. Bangladesh and India inherited these territorial problems in the form of enclaves in each other’s territories as well as a vaguely demarcated land boundary. The arbitrary division of the border between the two states resulted in India getting control over 112 enclaves and Bangladesh getting control over 32 enclaves. The division was made on the basis of the religious identities of the inhabitants. Further, the question of ownership of enclaves in the West Bengal–Bangladesh border and the Assam– Meghalaya–Bangladesh border remained unresolved. Another issue was regarding the question of ‘adversely held enclaves’, where an enclave legally belonged to one country but was in practice controlled by the other country. In 1974, Bangladesh and India signed a Land and Boundary Agreement to demarcate the border and prevent border conflicts. According to the agreement, these Indian and Bangladeshi enclaves were to be exchanged except for Berubari, Angorpota, and Dahagram
India's accusations on Islamic fundamentalism in Bangladesh: "Terrorism and the ‘war on terror’ are high on the agenda of each and every state, especially after the 11 September 2001 attacks in the United States." Western Media revealed that Bangladesh is evaporating with growing terrorist activities (Al-Qaeda..). India speculates that Bangladesh lets terrorist activities remain and no precautions are taken against such activities. Since political Islamism is growing in Bangladesh, India says this allows terrorist organizations to expand and India doesn't want terrorists in close Hindu borders.
citeseerx.ist.psu.edu. (n.d.). Download Limit Exceeded. [online] Available at: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1026.7349&rep=rep1&type=pdf [Accessed 14 Oct. 2020].
