Compensating variation versus change in value of goods consumed?
It wouldn't.  The value of consumer's endowments and the value after they trade should be the same.  i.e. Neither consumer becomes richer in terms of money.  They just allocate their money towards buying goods they value more.

Additionally, prices are a function of both consumers' demands while CV only depends on the demands of one consumer.  Consider the following example

In autarky consumer 1 has endowment, $e_1=(1,1)$ and utility 
$$u_1(x,y) = x + 2y $$ 
so $u_1(1,1)=3$
Now there is trade where consumer 2 has endowment, $e_2=(1,1)$ and utility 
$$u_2(x,y) = 2x + y $$
When the consumers trade the equilibrium price is $p_x = p_y$  and equilibrium consumption bundles are that consumer 1 gets $(0,2)$ and consumer 2 gets $(2,0)$.  The value of both consumers' endowments and equilibrium allocations is $2p_x$.  But in terms of compensating variation (with $x$ as the nominal good) consumer 1 would require 2 units of $x$ to move from free trade to autarky while consumer 2 would require 1 unit of $x$.