I strongly believe it won't matter if you marry a man who has a record of misdemeanor and sex offender, you can still get a job as a criminalist. You and your husband-to-be are different persons. You are not responsible of your man's offenses and mistakes.
I have searched over the internet and looked for job qualifications of crime scene investigators, and it doesn't include your issue. As long as you're physically, mentally, emotionally, and technically (which includes your educational background and trainings or certifications) fit to deal with crime investigations, then you will be qualified will definitely get a job.
However, if some agencies require special personal background check, then you can inquire with the agency you wanted to apply to.