What type of BITE is involved after installing a new display on an aircraft?
First, let's clarify the terms.
BITE (Built-In Test Equipment) is the hardware and software within a product that performs BIT (Built-In Test).
There are three common forms of BIT:

Power-on BIT (PBIT).  This is run every time the unit powers up. Depending on the unit function, some units have two variations of PBIT; A comprehensive check performed after a cold start, and an abbreviated check performed after a warm start.  Warm starts typically occur in-air after short power interruptions. The abbreviated PBIT is intended to get the unit back online as quickly as possible. PBIT only reports faults.
Continuous BIT (CBIT).  Generally implemented only in critical functions, CBIT provides a continuous (or near continuous, periodic) test. CBIT is a background test and like PBIT, only reports faults.
Initiated  BIT (IBIT).  This BIT is only run when it is initiated by the crew or maintenance personnel. It will typically take the longest time and be the most thorough BIT that can be performed. IBIT will, upon completion, provide a status report.

So to your question; when a new display is installed (or any other unit is replaced) and power is restored the unit will perform a PBIT and potentially begin CBIT (if it exists). After the unit is in normal operation, the technician that replaced it will normally perform an IBIT and verify the unit passed all tests.
It's important to remember that BIT is intended to identify hardware faults. If you are concerned with 'glitches' due to flipped bits (caused by single event upsets) or similar, that is typically addressed using integrity monitoring. This is more common than CBIT and it will only flag 'bad data' as it can't specifically identify what caused a fault. It only identifies that data is suspect as the primary computation and the monitor disagree. A hardware failure will generally result in the integrity monitor providing a continuous fault condition, but it's primary purpose is to identify 'soft' or intermittent faults.