Will buoyant force change if the container is accelerated up or down?

Archimedes' principle tells us that the upthrust on a body immersed in a fluid is equal to the weight of the fluid displaced, where the weight is the force given by $F = ma$ i.e. the mass of fluid displaced, $m$, multiplied by the acceleration, $a$, experienced by the fluid.

In this context there is no difference between gravitational acceleration and inertial acceleration - this is one example of Einstein's equivalence principle - so:

$$ a = a_{gravity} + a_{inertial} $$

And the upthrust is therefore:

$$ F = m (a_{gravity} + a_{inertial}) = V \rho (a_{gravity} + a_{inertial}) $$

as you said in your question.