NIT Trichy is one of the top 10 institutions in India for Engineering, NIT Patna makes it to the 56th position in the NIRF Rankings 2023 for Engineering.
Despite this, I would suggest you go for CSE from NIT Patna. Out of my experience of meeting different people, I would suggest you choose a better course over a better institution. CSE is in demand these days, while Civil might not be a good choice placement-wise.
A friend of mine pursued his B.Tech in Civil Engineering from a top NIT. He did not land a decent job. After that, he cracked GATE and went for his M.Tech in Civil Engineering from DTU. Even after his M.Tech, he landed a package of INR 5.5 LPA.
Civil Engineers from IIT Guwahati in my friend circle started preparing for UPSC because of not getting a job.
I believe that a course tag is permanent unless you switch your stream during your Masters. Therefore, I would suggest you take CSE from NIT Patna instead of Civil from NIT Trichy.