You don't "transfer". You get a new job.

I always advise new graduates: if you want to design, you need to be where design is happening.
To paraphrase Orson Welles, actors don't get discovered in Iowa.

Look for a new job. You are effectively entry-level with a few extra pluses. Be prepared to contract to gain experience.

Another word of advice - you may be short in some product design skills, but you are NOT short in desired design results! Absolute reliability. All engineers want their designs to work forever, but too many do not have the "talent" (read - hard experience) to properly anticipate the failures or shortcomings of their designs. They fail to think outside the box of their own creation and ask themselves "what could go wrong here?" At one time or another we've all said to ourselves, "If the guy that designed this thing had to operate it or work on it he would have done it differently." That's where your experience makes you stand out among your competition! You have seen the results of poor design practices. The design skills you need can be learned, but the reliability lessons you've learned come only from personal experience.

Learn to promote that. Accumulate some stories of specific events or projects from which you learned some valuable design lessons. I know I have several after a few decades.

You should not consider yourself behind your competition - but ahead!