If it has to come off, it's better to cut it now than wait for it to break. Growing it is just wasted energy for the tree.

I have a key lime in a pot and rooting some cuttings earlier this year. They all took, and citrus are said to be easy to root in general, so I say go for it. My cuttings were from the last three or four inches at the tip of a shoot. I took off all but the last few leaves, dipped them in a rooting hormone, and put them in plastic dixie cups (with holes in the bottom) filled with moist peat. The cups went into a larger plastic container with a thin layer of water at the bottom to keep the peat moist, then the whole thing went into a glass fishtank. Good luck!

I would cut off the whole branch then cut it into 4 equal parts and try to root all four.

If this is a grafted tree, the sprout may not be Key lime but may be from the rootstock. The way you described it seems to indicate a rootstock sprout. Check above the rogue limb for a callous where the graft union may have been made. If you bought the tree it is probably grafted. If the graft union is above the limb, cut it off and trash it. Just be aware your efforts to root this piece may be disappointing when it fruits and it isn't Key lime.

I've started several Key limes from seed. I plant the seed as soon as they are extracted from the fruit, while still wet with juice. I never dry them. They usually sprout very quickly.

Can i grow key lime through cuttings?