How much trouble does it cause my recruiter if I get married while in DEP (Delayed Entry Program)?

Don't worry about your recruiter. Any extra work this causes is part of his job. At most he will have to submit one extra form to put you both into the married couples program. If you wait until after training, as you say, you have a very good chance of spending a year in Korea while your husband spends a year in Montana. That isn't worth it! Good luck!