in my opinion, you should forget the voltmeter except as a reference of rough condition of the batteries state of charge,, and i do mean rough!
get a hydrometer, this is the only way to determine the state of charge of a lead acid cell with any certainty. learn how to use it properly, temp compensation as well. and get yourself a log chart started for each cell, so you can track the cells over time.
you will want to charge it with enough current to get it to boil a bit, so that the electrolyte doesn't stratify, boiling a bit will mix it well.
you want to charge it until the specific gravity will not rise any further, it would help if you contact the battery manufacture to find out what the specific gravity for that battery should be when fully charged. also ask them at what voltage they recommend you equalize the cells at. Don't get cute and think you know better than them, follow their recommendations and you will have much better luck keeping this battery up to its best condition.
using a voltmeter is fine for routine charging, but when you fully charge and especially when you equalize the cells a voltmeter is worthless (for determining state of charge), get a quality hydrometer!
at 1500amp/hrs, that battery will accept around 375amps charging current at between 50 and 80 percent depth of charge. so if you plan on working this battery you will need some way of putting some current into it or it will take forever to charge. 15amps might be fine for finishing off the charge, but will take a long time to charge the battery and will not get the battery to bubble which is needed to keep the electrolyte from stratifying.
and in my opinion, forget the damned desulfator!
in conclusion, contact the battery manufacture, find out at what voltages and currents they want it charged at, find out what equalization voltage they would like, and find out what the specific gravity should be on a fully charged cell in that battery.
follow their directions and you will have no need for a desulfator (those things drive me nuts!)
after going back and rereading your post, you state the battery has set for about a month and the voltage is around 12 volts,, that is basically a dead battery, so at 15 amps it is going to take several days to see the voltage come up. without a hydrometer it is going to be difficult to determine where you are with this battery.
hopefully this is helpful to you,
bob g