Can I get into CSE in IET Lucknow this year with a 2,371 rank in general quota taking into consideration that this year HBTI is through the Mains?

You have to wait till the final counseling takes place which is open for all. In the meantime, you should take admission in any branch you get there and wait for final counseling.

Taking into consideration that HBTI is now not taking admission through UPSEE the first option for students would be IET Lucknow.

Talking about the previous year's closing general rank in CSE was somewhat around 2600. So you can predict the answer to your question.

Check the cut off rank for IET Lucknow:

But you can also opt for EC, EE, EI, CE, and CHE as CS, IT, and ME may get filled earlier. In my opinion, you won't be getting any CSE seats on this rank in IET Lucknow.