I believe alternator B is not producing power, because either alternator B or voltage regulator B has failed. This is what’s I think is happening:
You start the engine with battery power via the Start Power switch or the Emergency Power Switch. Once running, the engine electrical power should be initially provided by alternator B, because alternator A does not come online until you reach 2500 (or maybe 2600) RPM for a few seconds. If alternator B does not produce power, and Start Power is removed before Alternator A is online, the engine will stop just as you describe.
The failure of alternator B (theory) is further supported by the fact that when you do get the engine running by starting above 2600 RPM, then the battery does not charge if either the start power or emergency battery switch is not closed. This makes sense because alternator B is normally charging the battery after alternator A connects to the engine (at which time alternator B is assigned to the airframe). With alternator B failed the battery will not charge. However, when you then close the emergency power switch you are connecting the airframe/battery to alternator A. Alternator A is not intended to charge the battery because it’s dedicated to the engine loads and normally isolated, but it will do that short term as long as power requirements are low.
Your successful starts (above 2600) are possible because at that RPM alternator A is almost immediately assigned to the engine, likely before Start Power is removed. Follow the prescribed procedure to test Stator B and Regulator B.