What about just digging it out and moving it to a better location? Looks like a nice sized Sabal seedling seeing as it's got a nice trunk developing on it.

Honestly, call up a nursery and ask them if they want a palm. Or put an add on Craigslist "Free palm to first person to dig it up!" Or just dig it up and put it on your curb, someone will grab it for you, more than likely.

Its a nice palm, I definitely think you should consider selling it on craigslist for a next to nothing price if they dig it up themselves. Looks like a Sabal Palmetto to me and a lot of people would really like a sabal palmetto that size since it's rare to find Sabal Palmettos that are anything smaller than tall palms with trunks.
You can dig it up yourself and move it or put it on the curb, if you get enough of the roots and the palm looks healthy, chances are someone will take it.
As for cutting back, when palm trees are cut below the growing point, they die, you probably didnt cut it down far enough and since the palm is so healthy, it clearly had no problem recovering! If you cut it down all the way to the ground, chances are it wouldn't come back. Palms are not one of those plants that will come back when cut to the ground.
-Alex

If its a sabal palm, then the roots will die back when dug up. And chances of survival are not good on sabals without a nice trunk. I would dig it and toss it.

Chadec, you should visit Florida some time. Digging up and transplanting sabals is a very common thing here. I-75 and I-95 always have a truck or two or a dozen loaded down with 12 - 24 sabals. I hate using my phone while driving, but if there's light traffic when I take my kids to school, I might snap a shot of a stand of 4-month planted sabals that are about 20 ft tall.

I have only seen the taller hurricane cut palms. I didn't think they did small ones.

I'll Take it. I'll Take good care of it. Depening if its a member of the sabal Palms.

I have only seen the taller hurricane cut palms. I didn't think they did small ones.

People seem to forget that palms are practically weeds in Florida! They will pop up anywhere a seed falls, whether you want a palm there or not. If it's too difficult to dig out, just keep cutting the leaves off at the base. It will run out of energy and die eventually.