According to the union, the dependence of the governors on the same advisors at BBC management had allowed the "flawed" plans to be approved without proper challenge.
Bectu says the proposal to sell the Technology subsidiary, which employees 1,4000 peoples, makes no sense and will cost the BBC more in technology costs than it currently pays.
In the open letter Grade is also asked to campaign against the "top-slicing" of the license fee, which would see some of BBC's income "creamed-off" to pay for public serving broadcasting on commercial channels.
He is also urged to pick up where Dyke left off in improving "goodwill towards the management". This comes as Dyke's replacement Channel 4 chief executive was named this week.