A few comments:
The buffer is declared here: https://github.com/arduino/Arduino/blob/master/hardware/arduino/avr/cores/arduino/HardwareSerial.h
Changing it for one serial TX port would require to make a seperate class for that port. That is not for beginners.
The Arduino TX never misses data.
When the TX buffer is full, the Serial.println() and Serial.write() wait until there is an open spot in the software buffer. That will slow down the sketch. Perhaps the sketch will become so slow that it no longer can be used. But all the data will be transmitted, always.
Therefor making that buffer larger does not solve your problem.
Changing to a Due will not solve the problem.
@Robin2, ha, ha, that is so funny.
Is the display running at 3.3V ? Is there a signal level problem between the 5V Arduino board and the display ?
Which display is it ?
Is it something like the Nextion displays ?
The display probably uses a microcontroller with hardware UART to read data. Perhaps the buffer in the display is too short, or the microcontroller too slow to read all the data.
What kind of cable do you use ? How long is it ? Can you make a photo of your project ?
Are those long strings really needed ?
How many bytes per seconds are transmitted to the display ?
Perhaps the problem is in the sketch. Can you make a minimal test sketch that shows the problem ? Can you show us the sketch ?
@KK959, we call this a XY-problem. You are looking for a solution, which has not to do with the problem.