You've identified the two most common solutions to this common problem. Which one you select depends on how you anchor base was designed and specified.
If you column base is a gravity only base (no moment or tension) welding may be an option. ASTM F1554, a common anchor rod material, is weldable in some of its varieties. Generally the lower strength materials (36 and 50ksi) are weldable. Higher strengths are too hard to weld. Drop forged rods are probably not weldable. I personally would not try to avoid welding rods designed for tension loads, either flexural or direct. If its a gravity base, the rods are really only there to aid errection and welding would be ok for this very light possible tension.
Anchor rod couplers are a better solution to rods where you need to develop tension.
If the rods are in tension and are weldable but couplers are not an option, you could splice the rod with a full pen weld which would need to be inspected.
My $0.02. Good luck.
