The reason a person can be compelled to testify after receiving a pardon is that they are no longer in jeopardy of incriminating themselves. http://time.com/4868418/donald-trump-presidential-pardons-backfire/
It would depend on the specific situation, but if you were in jeopardy of incriminating yourself in the state trial, you could plead the fifth, just like the Time.com article's hypothetical about testifying before congress.
You could bring up the pardon in the state court, if the judge allowed it, but that doesn't mean you are guilty just because you accept a pardon.
But Burdick was about a different issue: the ability to turn down a pardon. The language about imputing and confessing guilt was just an aside — what lawyers call dicta. The court meant that, as a practical matter, because pardons make people look guilty, a recipient might not want to accept one. But pardons have no formal, legal effect of declaring guilt.https://www.chicagotribune.com/news/opinion/commentary/ct-perspec-pardons-presidential-trump-nixon-ford-kardashian-0608-story.html
Whether it made a difference would be up to the jury, in that they would have to decide if it was relevant evidence that aided their deliberations.