This is a unfortunate difficult situation. Hopefully what Ashwin advised works (notebook computers display closed...with an external display attached)...this is called operating in "Clamshell mode".
Hopefully this works...and all is good. But if something should go awry...here are two additional ideas:
1. You don't need to reinstall the macOS fresh on the new 2TB SSD...just clone the current 256GB SSD to the new 2TB SSD using an app like Carbon Copy Cloner or Super Duper.
Probably will need an inexpensive external case for the 2TB SSD to do this. Additional benefit to this is...once the 2TB SSD is installed...you can insert the 256GB SSD into the external case...and repurpose it as an external storage device.
2. If you wanted to install a fresh copy of the macOS onto the new 2TB SSD...and if you have a 2nd Apple computer available to you...AND the macOS you want to install on the 2TB SSD is compatible with this 2nd Mac...you could again insert the new 2TB SSD into an external case...plug this into the 2nd Apple computer...and do the install this way.
Hopefully what Ashwin's advised above does the trick "clamshell mode"...then these suggestions unnecessary. Lol
Nick