You have encountered a defect in the Lotro game software. What you need to do is to use this form:
to submit a bug report to the Lotro Quality Assurance people or use the /bug command in game. Be sure to attach screen shots to your bug report along with adequate text to fully describe your problem with green dye. Make sure to include the race and gender of the character. Many items have race and gender specific artwork. They look fine on a Female Elf but horrid on a Dwarf.
The QA folks will forward it to an artist in the Lotro development organization. The artist will change the artwork for the Innkeeper's Dress so that the Green dye is visible Eventually the changes will be be made available to you in a future update to the game software.
The best way to see what an item will look like is to have a character of the same race and gender model the item for you in game. As you noticed there are sometimes differences between the preview and the actual look.
Unless stated otherwise, all content in this post is My Personal Opinion.