I realize that I struggle due to my underdeveloped EQ, but I would not be willing to give up any IQ points. Something in my brain is bellowing that I'd rather be smart than touchy-feely.

I doubt it is a trade off. Perhaps learning to apply one's IQ in another more efficient way when dealing with others would be a result of higher EQ. I have learned ways to interact so that I do not always seem as brusque or sarcastic at others. I draw the line at pandering to the hypersensitive. I told the one i see I did not want to turn into some crying, hugging, kumbaya singing, hand-holding, syruppy, over sharing, super sensitive whacko as a result of therapy. She indicated she was good but not a miracle worker. And then said something about middle ground being possible, blah blah blah.

Yes, I think there is a greater struggle for some than others, and that high intelligence/intellectual capabilities isn't necessarily going to make it easy to learn all aspects of EQ - and might in some ways work against one.
I think perhaps there is sometimes a deep, innate personality element involved, as in nature.....or a person was so deeply affected by environmental circumstances/upbringing, as in nurture, that the wiring, if you will, is very hard to change. That's not worded as precisely as I'd like to describe the mental image I have of what I mean.....
I think for myself I could have learned more EQ growing up when I was younger, my brain more malleable, my personality/traits/coping skills not so deeply embedded.....that I could have more easily learned emotional regulation/social graces, etc if I had not been so busy just surviving. I think my intelligence is what enabled me to develop some of the coping skills I needed to survive, my intelligence is one of the few definite cornerstones of my identity that I have always had.....but some things about EQ defy being learned purely through sheer intellectual ability, true. It takes somewhat of a different approach.....for me, it has been more of a spiritual one. My spirituality, for me, is something that has really helped me grow emotionally......
I too think that there are many different types of intelligence/abilities, many unique ways in which those are blended in each of us.....
For myself, I stink at numbers, I am not really a coolly logical (though I am analytical), mathematical, scientific type, but I am good with words, with abstract, conceptional thinking. I am really good at thinking about and parsing out my emotions ...... and I am good at sitting down and thinking about others' words/actions and parsing out their motives. I am good at the whole thinking thing ..... but I falter when it comes to the practical application of what I do with my perceptions, both accurate/inaccurate, how I should react accordingly. I can also analyze my emotions, but that doesn't mean I am so great at regulating them.....working on this.
Intelligence and wisdom are related, but also oh so different.....I am very smart, but not so wise; very intellectual, but not as versed in common sense, practical knowledge......
I choose to believe I am capable of learning, though, and that the effort is worth it. I can at least adapt/modify my behavior, even if I can't altogether change my personality/intelligence level and style/way of thinking or what goes on in the deepest parts of me. I accept that for me, certain aspects of social interactions/relationships are more of a struggle and I have to be more alert and think more what to do because it doesn't all come naturally/easily to me....