It is most likely the charging system. I would start by measuring voltage at the battery while the engine is idling (should be above 13v) and revved up around 5k (should be right about 14.7v). Chances are your charging system is barely producing 12v anywhere, which means either the regulator has quit, the stator is fried, or quite possibly, both have occurred.
I wouldn't bother taking the bike anywhere for a diagnosis. Most shops will want to get an old 2003 bike off their hands asap and will simply throw a battery at it until you keep coming back and pestering them. They may eventually charge you an arm and a leg for a crappy regulator they got from China that may or may not work. Bottom line, you need to get in there with a multimeter and diagnose it yourself. If you can spin a wrench, you can replace these items yourself with quality components.

 

here is a good charging system troubleshooting guide...different bike, but same principle:

The Definitive 675 Charging Diagnostic/Upgrade Thread

I know many of you have seen - and adopted - the R/R upgrade that I first authored on TriumphRat.net - Charging System Diagnostics - Rectifier/Regulator Upgrade Here is a tailored version of that specifically for the 675. Much of the content is derived from the original thread, so no apologies...

www.triumph675.net

btw, I've never heard of a battery needing to be "broken-in!" Just get yourself an inexpensive multimeter and go through the diagnosis systematically...there's not much else it could be other than the stator, regulator/rectifier, or battery...good luck

 

Definitely dive into the charging system. RR and stator tests

 

On my 2001 SV the battery died. For some reason the rectifier regulator also died. Had to replace that, used a GSXR 750 one. Also split the ground wire with a single wire back to the battery and into the full circuit.

 

The reason the R/R failed on your SV is because they all do on the first generation. 🙄

 

I would concur with the replies above, but would add, (and this is probably a bit of a thread hijack, for which I apologise), that a K3 SV650 with such low mileage is indeed a gem, but apart from whatever your charging system needs, due to the years of neglect that you mention, you will need to spent both time and money going through everything; change engine and brake fluids, take off brake calipers and check the pistons for corrosion, cleaning if possible, replacing if pitted. While the pistons are out check the seals and replace if at all suspect. Your tyres may well be unworn but they deteriorate with age, hardening and losing their ability to grip. They will most likely need to be replaced. There are plenty of videos on YouTube regarding the checking of the items I have mentioned and the work that will need to be done.