your answer is very dependant on how much Hussain is revered. That varies by each sect and I'm not sure you have chosen any specific sect.
Both Sunnis and Shias honor Hussain (in similar and different ways). Shias honor him more (they also have a 10 night mourn him in the first 10 days of Muharram), because he was also their Imam. I've know/seen Sunni brothers/sisters who visit and I know of some Sunnis who say he usurped against the caliphe Yazid...
Having that said Sunnis mostly believe that a Mahram must always be present. Shias don't believe such. Permission (of the guardian) is enough.Though most men accompany their wife/daughter/mother with other men because it's an extremely difficult trip.
About your country saying not travel:
I've traveled their 4-5 times after Saddam. Never had an issue. Many of my relatives went there in the past 3-12 months. Nothing happened. Though a very few have gone there and have been killed.Statistically less people going to the cities for pilgrimage in Iraq are killed than—people driving and having car accidents.
The major terror is not in Karbala, Najaf.
My advice: Have faith in Allah, go there with a group and be alert.