If you are a union member, you should consult your local. Otherwise, you will need to keep your tattoos covered. Perhaps you can think of other ways to keep your tattoos covered and convince your managers that they are acceptable means. Unless your tattoos have some First Amendment protection, which flowers likely do not, employers are entitled to enforce grooming and appearance standards. They just have to do that consistently without discriminating against any protected class.
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How many days is your trip? If you have vacation days, you should be able to use them unless approving your time off would cause the employer an undue hardship.
I would be interested in knowing what if any written policies the Company has on email/computer use and whether they have specifically reserved the right to monitor emails. I would also be interested in learning whether you know whether other employees' emails are routinely monitored and whether employees are always terminated when inappropriate emails are found.