Toss the old passport, get a clean one and go in (to Ireland and only Ireland) without mentioning the denied entry into the U.K. This is not a way to circumvent the denied entry, but rather a way to enjoy another country until the matter is resolved.

Separately write a letter to the British Consulate in the US and explain the circumstances of the denied entry and express that you are sorry and that you hope to have your record cleared and wish to go again to the U.K. on such and such a date.

Although the immigration officials are tasked with and empowered to guard the borders the decisions made at the border are reversible and erasable. The denied entry decision may have been right at the time for that official, but upon review they government may well find that you are not a risk and should be allowed in.

Been there, done all of this. It is something that can be handled legally and with a happy ending. A bit frustrating at the time (understatement of the world) but it is a situation where cooler heads prevail.