There are two ingredients in your question: First, statements invariant under forcing and second, large cardinals. The existence of certain large cardinals is a strengthening of ZFC.
Somewhat surprisingly, the known large cardinal axioms are essentially linearly ordered, i.e.,
given two such axioms, the consistency of one implies the consistency of the other.
Most set theorists believe that these large cardinal axioms are consistent.
Hence there is a natural direction to strengthen ZFC, namely by adding large cardinal axioms.
If we want to do mathematics, why not work in the strongest possible theory? Hence we assume
that large cardinals exist (like a proper class of Woodins).
Why are forcing invariant statements less intractable than others? Well,
simply because there is some hope that the statement can be decided in ZFC, or
in ZFC + large cardinals, which we believe in.
Moreover, if $\phi$ is invariant under forcing, we can actually use forcing to prove or disprove $\phi$.
An example is the Baumgartner-Hajnal partition theorem, a Ramsey like statement.
In the original proof it is shown that the statement is forcing invariant and follows from
Martin's Axiom. Since Martin's Axiom can be forced over every set-theoretic universe,
this shows that the Baumgartner-Hajnal statement is actually true in every model of set theory.
Now, of course there may be statements that are invariant under forcing, in particular statements about the natural numbers such as the Riemann hypothesis or P=NP, that might not be decidable in ZFC + large cardinals.
We currently have no method to prove independence results over ZFC other than forcing, inner models and consistency strength (the existence of a large cardinal cannot be proved in ZFC, because from the existence of the L.C. it follows that ZFC is consistent, but by the second incompleteness thm theorem this cannot be proved in ZFC (unless, of course, ZFC is inconsistent)).
If we are confronted with a statement that is forcing invariant, yet not decided by ZFC + LC, this statement is actually less tractable than others, because we currently may not have any way of proving its independence over ZFC + LC.
