If you ask me, I think you should opt for Mechanical at IIT Bhubaneswar rather than choosing ED at IITM. I am saying this not because I am a student of IIT Bhubaneswar but because of the high academic standards the institute maintains.
It is best to make a decision based on your own individual preferences and interests.
I also had the same dilemma and I ED at IIT Madras over other departments due to the following reasons:-
For more detail Refer - https://www.iitbbs.ac.in/index.php/home/placement-2023
https://internship.iitm.ac.in/downloads/P&I_Report_22-23.pdf
The choice should be based on your subject of interest. IIT Bhubaneswar is among the newly open IITs, whereas IIT Madras is an oldest and well-established institute.
IIT Madras offers dual degree programs in Engineering Design. It is a very practical course and offers a lot of opportunities. The students with a good GPA can get recruited by core companies. In 2019 placements, students of ED got placed in reputed companies with packages ranging between 7 LPA to 26 LPA. The placement rate of ED was 83%.
The department of Mechanical Engineering in IIT Bhubaneswar has good infrastructure and classrooms. It also has a product design lab, the place where all engineering design comes alive. A total of 89 companies visited the campus at IIT Bhubaneswar in the year 2019. Big core companies like Tata Steel, TCS, L&T, HP, Renault Nissan, Indian oil, and Maruti Suzuki recruited students from the ME department. In the year 2019, 83% of students of ME batch got placed in these companies with an average package of 11.43 LPA. The highest number of jobs was offered by the core sector. PSU like ISRO, IOCL, and NCCBM also participated in the campus placement. The department has several laboratories such as CAD & manufacturing, Robotics & Control, Nanomechanics, and Material science. Institute has also several mechanical related or robotics society which provides an excellent educational experience.