Welcome!
By the time you cut the barrel and rethread it, you are well in to the price of a shorter replacement barrel for an 870. You might also be out the price of new chokes unless the smith has the same thread cutter as Rem Chokes.
The various mail order shops will sell you a 20" inch Remington 870 barrel for around $150. Mossberg also sells 18.5" barrels for the 870 for about $100. I've seen these in stores for less than the price on Mossberg's website.
http://www.mossberg.com/acatalog/pricebarrels.htm
Cutting the barrel also makes it less desirable for clays because the longer barrel promotes follow through on the swing.