Is Bangladesh living in India or Chinese sphere of influence? If India, how much influence is India able to exert on Bangladesh?
These questions are a false choice. To begin, below I have the definition of what sphere of influence means. Strangely enough, I found the wikipedia definition to be the most comprehensive:
In the field of international relations, a sphere of influence (SOI) is a spatial region or concept division over which a state or organization has a level of cultural, economic, military, or political exclusivity, accommodating to the interests of powers outside the borders of the state that controls it.
The reason they are a false choice is that both China and India exert different aspects of what defines a sphere of influence. I will list off examples according to what aspect of "sphere of influence" that they follow. Here is how they both exert influence:
China's Sphere of Influence
Military: China makes 80% of Bangladesh's military equipment:
As for Sino-Bangladesh defence ties, Bangladesh’s dependence on Beijing has been long and stark. Dhaka [the capital of Bangladesh] imports 80% of its weapons from China and small arms are being made locally in collaborative ventures.
Economic: Bangladesh and China are part of the BCIM forum
Economic: China is Bangladesh's largest trading partner, but the partnership is tilted heavily in China's favor:
Bangladesh is third largest [read: now the first] trade partner of China in South Asia. But, the bilateral trade between them is highly tilted in favour of Beijing. The two-way trade stood at $1143 million in 2003-04. Bangladesh imported goods worth $1079 million against its export of $ 46 million to China.
There are other minor instances of China exerting it's influence on Bangladesh, which you may read here. Most of Bangladesh's policies in realtion to China may be traced to Ziaur Rahman.
India's Sphere of Influence
Military: India has increasing levels high level visits with Bangladesh's military chiefs.
Economic: India is Bangladesh's second largest importer and third largest overall trading partner according to the link on China's trading above.
Economic: India has given several loans to Bangladesh for developmental purposes.
Military: India signed two large defense agreements with Bangladesh in 2018:
Under the proposed bilateral defence agreement, India is willing to cooperate and collaborate with Bangladesh defence forces to set up manufacturing and service centres for the common defence platforms that both countries possess, informed persons familiar with the sector. India will equip the Bangladesh defence forces to meet their demand of expert training, along with technical and logistic support. This will also enable Bangladesh to attain self-sufficiency in defence manufacturing the long run.
Cultural: Much of Bangladesh's music descends directly from Indian instruments and tradition.
Political: India has a shoot-to-kill policy as a means to deal with the Bangladesh-India border issue. Multiple Bangladeshi governments have expressed concerns that they are undermined by India's policy:
Consecutive governments in Dhaka have argued that the fence is part of a wider aggressive stance adopted by India to maintain its position as the regional superpower and the rhetoric is an attempt to undermine Bangladesh. They declare that the Bangladeshi border guards, the Bangladesh Rifles (BDR) are frequently thwarting smugglers and would-be immigrants in the frontier areas.