When I create a Performance and add Drums to Part 4, the drums start repeating with each note I play from the Piano in Part 1 of the Performance Mode
Most likely you have selected a Drum Arpeggio that features an Accent articulation. Typically, the accent is a Kick + Crash "punch" when you exceed the ACCENT VELOCITY THRESHOLD (play above the threshold)
Among the arpeggio edit parameters is "AccntVelTh" which is the velocity at which you can have the punch articulated. If set too low you may get this accent when you don't want it.
Press [ARP EDIT]
Press [F5] PLAY FX
Select the Part containing the drum kit.
There are two pages of parameters; use the [>]/[<] arrows to move between them
"AccntVelTh" should be set high enough so you only trigger it at or above that value.. Or you can set it to Off.
The other possibility is that you have not set the Arp to "HOLD = ON" the ARP EDIT > [F3] MAIN screen. In addition to each Part having its own Arp On/Off Switch, each Arp can be set Hold. If set Off this would cause the arp phrase to stop any time you were not in contact with a depressed key or the sustain pedal.