It's a grey area for you, but although it makes common sense not to tax, it is illegal not only to drive without tax but also illegal not to display a valid disc. To be fair it would be almost impossible to legislate for your brief grey area need in the state. The correct course of action is to tax and then apply for a rebate when you re-register the car in your new country.
Not sure about driving across UK untaxed (I'd guess they would not bother with the paperwork, if there is such a common EU rule) but I'd guess not having a valid disc might draw attention to you at borders when you are really just going about your business innocently.
I'd check with a Garda station and then it's up to you if you want the risk, just know that the law is usually not grey.