You may find that you will have better chances at a
small company, where roles are not so rigidly defined.
You might get in with your present experience in test,
highlighting the ability to get things done in the "real
world" and the things you have had to design in order
to perform the testing ( load boards, evaluation circuits,
etc.) and a stated desire to do more designing in the
new job.
Large companies are full of people whose job is to
say "no". You need to find a company that says,
"when can you start?".
I was fortunate and determined, I turned down a
few job offers in product engineering before I got
my first engineering job, in design. I went into an
area which was somewhat unpopular at the time,
but design nonetheless. Over time I was able to
migrate to more interesting specialization.
Unlike when I first got into semiconductors, there is
a very large pool of fabless and design-for-hire shops
out there now. These always seem to lack the test
and product support strength they need. Maybe one
of these will take you on for your present skills in
exchange for helping you develop the ones you want.
Make a deal.