Ouch, kathy ..so sorry to hear of your pain. I have tailbone/coccyx pain (that runs down my leg) with pelvic issues off and on ....it's terrible at times. I barely do any core work on my back and things resembling boat pose (like paddling) are out 85% of the time. I am enjoying ICE a lot but I hardly do the core workouts on it as it is mainly "crunchy"/on the back.(of course, most people love it and I am NOT dissing ICE as I am using it often and think it is a treasure!!

) Lying down core work isn't great for me due to neck issues and pelvic issues which cause very tight psoas and coccyx discomfort.There are other ways to work and condition the core area. Kettle Bell/ unilateral dumbbell work being one I like. (when your form is excellent, standing core work can be better than a lying down routine, IMHO).
Have you put any heat on it? (an electric heating pad helps me immensely!) Also, if you feel you have repetitive injury, I would highly recommend a couple of professional massages. Tailbones can be tricky as sometimes the pain is referring from somewhere else particularly along the spine, so if these things do not work out for you, I would suggest some radiology.
Take care. I honestly feel this boat/paddling position simply does not work for every body. You could however, try putting a stability ball against a wall, leaning on it and trying it (AFTER you have healed up).
(Obviously I am not Cathe

....I could relate to what you were feeling!!) All the best.