I don’t have Big Sur, but the idea that a computer forgets about a device once it is being paired to another sounds very strange to me — a lot of devices are meant to support multiple concurrent pairings.
So either Big Sur tries to be too clever in some obscure way, or it’s your trackpad which actively says the old computer good bye when being paired anew. Which specific trackpad model are you using? And does this happen with other Bluetooth devices, too?