It's possible to travel either with Irish Ferries or Stena Line from Holyhead to Dublin.
In case it's of help it's possible for the sailrail ticket to commence from any local station too eg Aigburth/Kirkdale/Crosby/Hunts Cross.
Normal route would be a Merseyrail train to Chester and catch a Holyhead train there.
Birth cert should be fine as id, a family member travelled with me at short notice just over a year ago and hadn't passport up to date. Birth cert was accepted without any problem at the check at Holyhead.
Perhaps the 13.50 hrs Stena Line ferry from Holyhead arriving Dublin 1705. A bus will bring you from the ferry terminal door to Heuston Station in good time for the 18.15 train. Arrive in Claremorris at 20.39.
Www.nationalrail.co.uk will show train times from Lime Street/Local station to Holyhead. Will try and post suggested times later. Hope this info is of some help. Plenty on the ship to keep the kids amused. Train journey along the coast from Chester to Holyhead is scenic, Conwy Castle, Puffin Island, Menai straits