Place an ammeter in between battery and wiring. Remove fuses one by one till the ammeter doesn't show a draw. That will indicate which circuit is the culprit. Then you'll have to remove the different items that are on the circuit to find the one drawing the current.
Any wires coming off the battery that aren't factory? They may have their own fuse (inline) some where. You could have a wire that has been chaffed and is 'leaking'. I would chase the "hot" wires to insure that they aren't compromised somewhere. A lot of these bikes were 'modded' electrically and you might find abandoned wiring mods. Best of luck...
I would suggest you get a factory Electrical Troubleshooting Manual which will have the circuit diagrams. These are very complicated bikes and after 37years alot of crap connections could have been made.
I would remove any aftermarket electrical connections/additions from wherever - get back to the OEM wiring design. The OEM wiring harness is designed much like a house, the wiring is to the minimum size to safely operate the motorcycle as it was designed and how it came from the factory, after which we get our hands on the motorcycle and do all sorts of wondrous things to the electrical system.
ditto returning to stock. the electrical system can barely fulfill the needs of the stock bike when rpms are over 2600 rpm. it does what it was designed to do, and does it fairly well, but lots of folks think it should look like a lit up semi and have a stereo amp and other stuff. then there is the less than skilled "mechanic " with splices, wire nuts, etc.
Lots of good recommendations here. A follow on for when you find and correct the voltage leak.
The issue is that the electrical system on these motorcycles is time tested to perform well as designed. 30/40 year old electrical systems regardless of the platform need a lot of maintenance. Dirty/crud filled connectors, and poor grounds is our nemesis.
How much juice is available - probably not much after the electrical system design is taken care of. Most people do not consider this. Your 1200 has approximately 350 watts of power at approximately 5500 RPM - you do not have 350 watts of power available at the lower RPMs. Sliding scale from this to idle at ~1000 RPM. Electrical system needs say 10 to 18 amps for normal operation. Have about 10 amps left over for additions and this is at 5500 RPM.
The 350 watt "alternator" has an amp range of 25 to 29 amps max at ~5500 RPM. This is dependent on whether you want to use 12 VDC as the constant or the RR reference point of 14.2 VDC - your choice. We all know that the stator does not generate enough power at idle to operate the electrical system, and when this happens the battery is being used to supplement the requirement.
This has to be considered when you load up the electrical system. Since the "alternator" design is RPM dependent, you cannot make any more power at a given RPM, and this can be taxing on the charging system. If you do load up the electrical system, make sure you always have a good battery installed.
A good battery is needed because it has three requirements. #1 - start the engine. #2 - absorb voltage spikes and such when operating. #3 - supplement the electrical system requirements when the alternator (3 part consisting of RR, stator, rotor) cannot produce enough power to meet the requirement. After a start, the battery is quickly charged to 100% ( never gets there), and receives a trickle charge afterwords. Most of the electrical power is diverted to system operation. This is the same for issue #3. Once the battery is charged, it is a "passenger" in the electrical system.
The battery can be AGM or the newer lithium-ion type. Both are good. The lithium-ion works well in the cold and warmer climates.
This is a good site to visit - Alberta Lithium Battery Company: Alberta Lithium Battery Company. This company uses the lithium-ion batteries in ATVs, snowmobiles and other applications successfully - bought mine from this company. I will mention that if you have to do continuous testing of the engine - start/stops, use an AGM or garden style battery. Lithium-ion batteries do not do so well in a testing environment.
If the battery is starting to fail (internals degrading), it will draw an excessive amount of power from the system, and makes for poor engine operation. This is especially true for the FI models, BTDT.
The RR has a voltage reference point of approximately 14.2 VDC, same as an automobile. This can vary depending on where you take the reading from. The issue is not so much the reading, but the consistency of the reading at the connection. If it remains the same when operating, good, only be concerned when it starts to change - then go looking.
A good upgrade is to replace the older shunt RR with a newer model that has better internal electronic components, or better yet is to change to a series RR. Shunt RR regulates the electrical system voltage by syphoning power off to ground or not. A series RR stops/starts electrical flow to the system. There is no such RR as a MOSFET RR - series or shunt. A MOSFET is an internal electronic component used in shunt and series RRs, and for other electrical applications - been around since '59. The name has caught on, much like every snowmobile is a Skidoo. Suffice it to say, make sure you are getting the RR you want that does what you expect it to do - series or shunt. This is good article on the newer shunt RR regarding this issue: RM Stator MOSFET Regulator Upgrade The RM Stator company has other electrical components such as coils for the 1200 at reasonable prices. Have not bought from them, but like that this company is in Canada.
A series RR used by the Polaris ATV world is a good series RR option. Older internal electronic technology, but still a good series RR. Smaller installation footprint than the larger ones you get from Roadster Cycle such as the SH847.
The series RR that I have found to be most used are the SH847 50 amp, Compu-Fire 55402 40 amp, and the Polaris RR (have used the SH847 and Computer-Fire). These as well as the newer shunt RRs only have 5 wires; 3 stator and power/ground wires. These sense the system voltage through the power wire - no sensing wire required.
The OEM generally has 3 stator, 2 grounds, 2 power, and a sensing wire. The OEM RR ground and power wires merge into a single ground and power wire respectively just after disappearing into the wiring harness.
If you do upgrade, make sure you protect the original wiring and connectors as some of the wires in the original connector(s) are live at all times.
Once you get the issue corrected, and want to add additional electrical items, I would recommend a separate accessory fuse box, combined power and ground connections, that is powered through an automotive relay. I have one installed in the rear trunk - not a lot of space available for extra items on these bikes. I would connect the power wire to the relay at the "B"(battery) connection of the starter relay, this is how the Computerized Fuel System (CFI) on the FI models is done. The negative connection can go directly to the battery. Need to find a switched 12 VDC wire to trigger the relay. Install all non-OEM items to this, power and ground. This allows you to easily troubleshoot any electrical fault by being able to disconnect this fuse box from the system and determine what the issue is without wondering if it is from an original install, or did I do something when I installed the extras.
Good luck, trust that I am not highjacking your thread, and hope you resolve your issue soon. Have a good riding season.