I would not suggest that you "clean" the roots of the existing medium. This will only damage the roots and is not necessary. Do you suspect something is wrong with the soil (fungal, etc.?) I also wouldn't grow citrus in coconut husk. (Isn't that medium mostly for orchids and bromeliads?) I have my citrus growing in a standard potting soil amended with an extra helping of peat moss and coarse sand. Aim for a mix that is acidic and promotes adequate drainage. Citrus are not the best subjects for transplanting to begin with. Their roots are relatively fragile and easily damaged and they seem to resent being moved. (You might want to wait until March to transplant.) Also, you don't want a pot that is excessively large for the size of the plant/tree. They like is slightly on the snug side--that's my impression from years of growing them in containers anyway. Hope this was helpful! Good luck!

I repotted my meyer lemon last spring. I removed all of the old soil when I repotted into a faster draining soil (similar to bark / turface / granite) - see container soils. I didn't have any problems with the repot. I also repotted into a larger container, but that is only because the mix I'm using drains so fast.
You may want to read up a little more before choosing to repot into coconut husk / coir.

Having two moderately different soil types in a container really isn't that big of a deal in the sense of one type closer to the trunk and one farther away. I pull the plants out, shake or jar them a bit to shake the soil off the outer couple inches of roots. This is to have roots protruding well into the new soil and to uncoil the roots a bit. Put some new type soil in the bottom of the pot and to surround the old soil and root ball. Works fine for me. If a lot of the old soil falls off, so be it, so much the better. Then I usually mix it with the new soil before replacing.

I agree with zubaby, I also repotted my Dwarf Meyer Lemon in Al's Gritty Mix(the link she provided). I also removed all the old soil and repotted in that awesome, fast draining soil...the lemon is now in a 14" pot. I purchased it in a one gallon pot appx. 3 years ago. Since I have used this Potting Mix, it has done very well! I had no problems with the roots when I removed the old soil. For me, I would say two different types of soil can be damaging...one that stays moist all the time, while the other stays drier. Just my experience and opinion.

Just use caution, at the time of year when repotting, I live in Fl, so repotting almost any time of the year is fine for me.