That is totally against how the institutions are built today.
NAL is a R&D lab focusing on civil aviation. ADE is lab part of DRDO working on military aviation but ADA is a separate autonomous programme-management agency. There is no overlap at the project level. ADA only takes by high level critical projects like LCA and AMCA. ADE do a whole lot just like any other DRDO lab.
ARDC is part of a corporate entity called HAL. It is up to them how to organize. It is impossible to merge with fully government labs.
If anything we should be encouraging private entities like TASL to make their own version of R&D lab and duplicate efforts. We should be concerned about competition not duplication of efforts. For example, IAF need a new trainer. They define what they want and industry (HAL and TASL) work on their design parallel then the IAF chooses best design for them and give them full contract. We have the scale, what we lack is private industry for competition as the entire space is new for them.
We almost have the perfect structure for this because of the organisation like DRDO. They can be the place industry work with to design and make use of testing infrastructure. You can see this happening in every vertical. Pinaka with L&T, Solar. ATAGS with BF and TATA. FICV with BF, L&T and TATA etc. It can happen for the land systems more easily because the industry is way more mature and confident in automobiles. Thats not the case of civil aviation. Do we complain when new rocket companies comes up ? Its the same script. So, let there be duplication.